Grade 11 : Physics - Thermodynamics (Set 3)

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Grade 11 : Physics - Thermodynamics (Set 3)— Questions & Detailed Solutions

A constant-volume gas thermometer measures a pressure of $1.20 \times 10^5 \text{ Pa}$ at the triple point of water (273.16 K).
What temperature corresponds to a measured pressure of $1.50 \times 10^5 \text{ Pa}$ with this thermometer?

(A)

341.45 K

(B)

328.50 K

(C)

350.20 K

(D)

298.15 K

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: For a constant-volume gas thermometer, pressure is directly proportional to absolute temperature: $T = T_{tp} \left(\frac{P}{P_{tp}}\right)$.
$$T = 273.16 \times \frac{1.50 \times 10^5}{1.20 \times 10^5} = 273.16 \times 1.25 = 341.45 \text{ K}$$

A sample of ideal gas undergoes an isobaric expansion at a constant pressure of $2.0 \times 10^5 \text{ Pa}$ from an initial volume of $0.010 m^3$ to a final volume of $0.025 m^3$.
If 9000 J of heat is supplied to the gas during this process, what is the change in internal energy $\Delta U$?

(A)

3000 J

(B)

6000 J

(C)

9000 J

(D)

12000 J

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Work done in an isobaric process is $W = P \Delta V = 2.0 \times 10^5 \times (0.025 - 0.010) = 3000 \text{ J}$.
By the First Law of Thermodynamics, $\Delta U = Q - W = 9000 \text{ J} - 3000 \text{ J} = 6000 \text{ J}$.

Two moles of an ideal gas maintained at a constant temperature of 300 K expand isothermally from $V_1 = 2 \text{ L}$ to $V_2 = 2e^2 \text{ L}$ (where $e \approx 2.718$). Use $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
Calculate the work done by the gas during this isothermal expansion.

(A)

4986 J

(B)

9972 J

(C)

2493 J

(D)

14958 J

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Work done in an isothermal process is $W = n R T \ln\left(\frac{V_2}{V_1}\right)$.
$$W = 2 \times 8.31 \times 300 \times \ln(e^2) = 4986 \times 2 = 9972 \text{ J}$$

A monoatomic ideal gas ($\gamma = 5/3$) initially at 300 K is compressed adiabatically until its volume is reduced to $\frac{1}{8}$ of its initial volume.
What is the final temperature of the gas?

(A)

600 K

(B)

900 K

(C)

1200 K

(D)

2400 K

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: For an adiabatic process, $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$.
$$T_2 = T_1 \left(\frac{V_1}{V_2}\right)^{\gamma-1} = 300 \times (8)^{(5/3 - 1)} = 300 \times 8^{2/3} = 300 \times 4 = 1200 \text{ K}$$

Cyclic process $A \to B \to C \to A$ on a P-V diagram:
State A: $(1 m^3, 100 \text{ kPa})$
State B: $(4 m^3, 100 \text{ kPa})$
State C: $(4 m^3, 400 \text{ kPa})$
Path $C \to A$ is a straight line.
What is the net work done by the gas during one complete cycle?

(A)

450 kJ

(B)

900 kJ

(C)

300 kJ

(D)

150 kJ

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: Net work done equals the enclosed area of the triangle in clockwise order:
$$W_{net} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times (4 - 1) \text{ m}^3 \times (400 - 100) \text{ kPa} = \frac{1}{2} \times 3 \times 300 = 450 \text{ kJ}$$

For a gas with ratio of heat capacities $\gamma = 1.40$, the universal gas constant is $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
What is the molar heat capacity at constant volume $C_v$?

(A)

20.78 J/(mol·K)

(B)

29.09 J/(mol·K)

(C)

12.47 J/(mol·K)

(D)

31.16 J/(mol·K)

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: Using $C_p - C_v = R$ and $\gamma = C_p / C_v$, we have $C_v = \frac{R}{\gamma - 1}$.
$$C_v = \frac{8.31}{1.40 - 1} = \frac{8.31}{0.40} = 20.775 \approx 20.78 \text{ J/(mol}\cdot\text{K)}$$

1 mole of a rigid diatomic ideal gas ($C_v = \frac{5}{2}R$) is heated at constant volume from 300 K to 500 K. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
How much heat $Q$ is transferred to the gas?

(A)

2493 J

(B)

4155 J

(C)

5817 J

(D)

1662 J

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: At constant volume, work $W = 0$, so $Q = n C_v \Delta T$.
$$Q = 1 \times \left(\frac{5}{2} \times 8.31\right) \times (500 - 300) = 2.5 \times 8.31 \times 200 = 4155 \text{ J}$$

Consider 2 moles of a non-linear triatomic ideal gas with 6 degrees of freedom ($f = 6$) at a temperature of 400 K. ($R = 8.31 \text{ J/(mol}\cdot\text{K)}$).
Calculate the total internal energy $U$ of this gas sample.

(A)

9.97 kJ

(B)

19.94 kJ

(C)

29.92 kJ

(D)

13.30 kJ

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Internal energy of ideal gas is $U = \frac{f}{2} n R T$.
$$U = \frac{6}{2} \times 2 \times 8.31 \times 400 = 6 \times 8.31 \times 400 = 19944 \text{ J} \approx 19.94 \text{ kJ}$$

A Carnot engine operates between a high-temperature reservoir at $227^\circ\text{C}$ and a low-temperature reservoir at $27^\circ\text{C}$.
What is the maximum theoretical efficiency $\eta$ of this engine?

(A)

20%

(B)

40%

(C)

60%

(D)

80%

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Convert temperatures to Kelvin: $T_H = 227 + 273 = 500 \text{ K}$, $T_C = 27 + 273 = 300 \text{ K}$.
$$\eta = 1 - \frac{T_C}{T_H} = 1 - \frac{300}{500} = 0.40 \text{ or } 40\%$$

Q10

A refrigerator operates on a reversed Carnot cycle between $-23^\circ\text{C}$ and $27^\circ\text{C}$.
What is the coefficient of performance (COP) of this refrigerator?

(A)

2.5

(B)

4.0

(C)

5.0

(D)

6.0

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: Convert temperatures: $T_C = -23 + 273 = 250 \text{ K}$, $T_H = 27 + 273 = 300 \text{ K}$.
$$\text{COP} = \beta = \frac{T_C}{T_H - T_C} = \frac{250}{300 - 250} = \frac{250}{50} = 5.0$$

Q11

A heat engine absorbs 2000 J of thermal energy from a high-temperature source during each cycle and performs 600 J of mechanical work.
Find the heat rejected to the cold reservoir and the thermal efficiency.

(A)

1400 J heat rejected, 30% efficiency

(B)

1400 J heat rejected, 40% efficiency

(C)

600 J heat rejected, 30% efficiency

(D)

800 J heat rejected, 70% efficiency

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: Heat rejected $Q_C = Q_H - W = 2000 \text{ J} - 600 \text{ J} = 1400 \text{ J}$.
Thermal efficiency $\eta = \frac{W}{Q_H} = \frac{600}{2000} = 0.30 \text{ or } 30%$.

Q12

An ideal gas expands freely into an evacuated insulated chamber.
Which set of thermodynamic parameters correctly describes this free expansion process?

(A)

Q = 0, W = 0, ΔU = 0, ΔT = 0

(B)

Q > 0, W = 0, ΔU > 0, ΔT > 0

(C)

Q = 0, W > 0, ΔU < 0, ΔT < 0

(D)

Q < 0, W = 0, ΔU < 0, ΔT < 0

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: Insulation implies $Q = 0$. Expansion into vacuum against zero external pressure means $W = 0$. By the First Law, $\Delta U = Q - W = 0$, and for an ideal gas $\Delta U \propto \Delta T = 0$.

Q13

A monoatomic ideal gas ($C_v = \frac{3}{2}R$) undergoes a polytropic process following $P V^2 = \text{constant}$.
What is the molar heat capacity $C$ of the gas during this process?

(A)

0.5 R

(B)

2.5 R

(C)

-0.5 R

(D)

1.5 R

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: For a polytropic process $P V^n = \text{const}$, $C = C_v + \frac{R}{1 - n}$.
Here $n = 2$, so $$C = \frac{3}{2}R + \frac{R}{1 - 2} = \frac{3}{2}R - R = \frac{1}{2}R$$

Q14

Consider an isothermal process and an adiabatic process passing through the same point $(P_0, V_0)$ on a P-V diagram for an ideal gas with ratio of heat capacities $\gamma$.
What is the ratio of the absolute slope of the adiabatic curve to that of the isothermal curve at this point?

(A)

(B)

(C)

1/γ

(D)

γ - 1

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Slope of isothermal curve: $\left|\frac{dP}{dV}\right|_{iso} = \frac{P}{V}$.
Slope of adiabatic curve: $\left|\frac{dP}{dV}\right|_{adi} = \gamma \frac{P}{V}$.
Ratio $= \frac{\gamma (P/V)}{P/V} = \gamma$.

Q15

A gas mixture consists of 1 mole of Helium ($C_v = \frac{3}{2}R$) and 1 mole of Oxygen ($C_v = \frac{5}{2}R$).
What is the effective molar heat capacity at constant volume $C_{v,\text{mix}}$ of this mixture?

(A)

1.5 R

(B)

2.0 R

(C)

2.5 R

(D)

3.0 R

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Effective molar heat capacity is given by $C_{v,\text{mix}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2}$.
$$C_{v,\text{mix}} = \frac{1 \times \frac{3}{2}R + 1 \times \frac{5}{2}R}{1 + 1} = \frac{4R}{2} = 2.0 R$$

Q16

Cyclic process for an ideal gas:
1. Isobaric expansion from $(P_0, V_0)$ to $(P_0, 3V_0)$
2. Isochoric pressure drop from $(P_0, 3V_0)$ to $(P_0/2, 3V_0)$
3. Straight line return path from $(P_0/2, 3V_0)$ back to $(P_0, V_0)$.
What is the net work done by the gas in one cycle?

(A)

(1/4) P₀V₀

(B)

(1/2) P₀V₀

(C)

P₀V₀

(D)

2 P₀V₀

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: The path forms a clockwise triangle on the P-V diagram.
$$W_{net} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times (3V_0 - V_0) \times \left(P_0 - \frac{P_0}{2}\right) = \frac{1}{2} \times 2V_0 \times \frac{P_0}{2} = \frac{1}{2} P_0 V_0$$

Q17

2 moles of an ideal gas undergo an isothermal expansion at 400 K from volume $V$ to volume $e^3 V$. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
Calculate the change in entropy $\Delta S$ of the gas.

(A)

16.62 J/K

(B)

24.93 J/K

(C)

49.86 J/K

(D)

99.72 J/K

View Correct Answer & Detailed Solution

✓ Correct Option: (C)

Explanation: For an isothermal process, $\Delta S = \frac{Q}{T} = n R \ln\left(\frac{V_2}{V_1}\right)$.
$$\Delta S = 2 \times 8.31 \times \ln(e^3) = 2 \times 8.31 \times 3 = 49.86 \text{ J/K}$$

Q18

During an adiabatic process involving 1 mole of gas ($\gamma = 1.4$), the state changes from $(P_1 = 4.0 \times 10^5 \text{ Pa}, V_1 = 1.0 \times 10^{-3} m^3)$ to $(P_2 = 1.0 \times 10^5 \text{ Pa}, V_2 = 2.5 \times 10^{-3} m^3)$.
What is the work done by the gas during this process?

(A)

250 J

(B)

375 J

(C)

500 J

(D)

625 J

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Work done in an adiabatic process is $W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}$.
$$W = \frac{(4.0 \times 10^5 \times 1.0 \times 10^{-3}) - (1.0 \times 10^5 \times 2.5 \times 10^{-3})}{1.4 - 1} = \frac{400 - 250}{0.4} = \frac{150}{0.4} = 375 \text{ J}$$

Q19

A monoatomic ideal gas ($\gamma = 5/3$) expands isobarically when heat is added to it.
What fraction of the total heat energy supplied is converted into external mechanical work?

(A)

20%

(B)

40%

(C)

60%

(D)

66.7%

View Correct Answer & Detailed Solution

✓ Correct Option: (B)

Explanation: Fraction of heat converted into work in isobaric expansion is $\frac{W}{Q} = \frac{P \Delta V}{n C_p \Delta T} = \frac{R}{C_p} = 1 - \frac{1}{\gamma}$.
$$\frac{W}{Q} = 1 - \frac{1}{5/3} = 1 - \frac{3}{5} = \frac{2}{5} = 0.40 \text{ or } 40\%$$

Q20

A Carnot engine operating with a thermal efficiency of 25% produces 500 J of mechanical work per cycle.
How much heat is absorbed from the hot reservoir and exhausted to the cold reservoir each cycle?

(A)

Q_H = 2000 J, Q_C = 1500 J

(B)

Q_H = 1500 J, Q_C = 1000 J

(C)

Q_H = 2500 J, Q_C = 2000 J

(D)

Q_H = 2000 J, Q_C = 500 J

View Correct Answer & Detailed Solution

✓ Correct Option: (A)

Explanation: Efficiency $\eta = \frac{W}{Q_H} \implies 0.25 = \frac{500}{Q_H} \implies Q_H = 2000 \text{ J}$.
Heat exhausted $Q_C = Q_H - W = 2000 - 500 = 1500 \text{ J}$.

Grade 11 : Physics - Thermodynamics (Set 3)

Grade 11 : Physics - Thermodynamics (Set 3)— Questions & Detailed Solutions

A constant-volume gas thermometer measures a pressure of $1.20 \times 10^5 \text{ Pa}$ at the triple point of water (273.16 K).
What temperature corresponds to a measured pressure of $1.50 \times 10^5 \text{ Pa}$ with this thermometer?

A sample of ideal gas undergoes an isobaric expansion at a constant pressure of $2.0 \times 10^5 \text{ Pa}$ from an initial volume of $0.010 m^3$ to a final volume of $0.025 m^3$.
If 9000 J of heat is supplied to the gas during this process, what is the change in internal energy $\Delta U$?

Two moles of an ideal gas maintained at a constant temperature of 300 K expand isothermally from $V_1 = 2 \text{ L}$ to $V_2 = 2e^2 \text{ L}$ (where $e \approx 2.718$). Use $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
Calculate the work done by the gas during this isothermal expansion.

A monoatomic ideal gas ($\gamma = 5/3$) initially at 300 K is compressed adiabatically until its volume is reduced to $\frac{1}{8}$ of its initial volume.
What is the final temperature of the gas?

Cyclic process $A \to B \to C \to A$ on a P-V diagram:
State A: $(1 m^3, 100 \text{ kPa})$
State B: $(4 m^3, 100 \text{ kPa})$
State C: $(4 m^3, 400 \text{ kPa})$
Path $C \to A$ is a straight line.
What is the net work done by the gas during one complete cycle?

For a gas with ratio of heat capacities $\gamma = 1.40$, the universal gas constant is $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
What is the molar heat capacity at constant volume $C_v$?

1 mole of a rigid diatomic ideal gas ($C_v = \frac{5}{2}R$) is heated at constant volume from 300 K to 500 K. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
How much heat $Q$ is transferred to the gas?

Consider 2 moles of a non-linear triatomic ideal gas with 6 degrees of freedom ($f = 6$) at a temperature of 400 K. ($R = 8.31 \text{ J/(mol}\cdot\text{K)}$).
Calculate the total internal energy $U$ of this gas sample.

A Carnot engine operates between a high-temperature reservoir at $227^\circ\text{C}$ and a low-temperature reservoir at $27^\circ\text{C}$.
What is the maximum theoretical efficiency $\eta$ of this engine?

A refrigerator operates on a reversed Carnot cycle between $-23^\circ\text{C}$ and $27^\circ\text{C}$.
What is the coefficient of performance (COP) of this refrigerator?

A heat engine absorbs 2000 J of thermal energy from a high-temperature source during each cycle and performs 600 J of mechanical work.
Find the heat rejected to the cold reservoir and the thermal efficiency.

An ideal gas expands freely into an evacuated insulated chamber.
Which set of thermodynamic parameters correctly describes this free expansion process?

A monoatomic ideal gas ($C_v = \frac{3}{2}R$) undergoes a polytropic process following $P V^2 = \text{constant}$.
What is the molar heat capacity $C$ of the gas during this process?

Consider an isothermal process and an adiabatic process passing through the same point $(P_0, V_0)$ on a P-V diagram for an ideal gas with ratio of heat capacities $\gamma$.
What is the ratio of the absolute slope of the adiabatic curve to that of the isothermal curve at this point?

A gas mixture consists of 1 mole of Helium ($C_v = \frac{3}{2}R$) and 1 mole of Oxygen ($C_v = \frac{5}{2}R$).
What is the effective molar heat capacity at constant volume $C_{v,\text{mix}}$ of this mixture?

Cyclic process for an ideal gas:
1. Isobaric expansion from $(P_0, V_0)$ to $(P_0, 3V_0)$
2. Isochoric pressure drop from $(P_0, 3V_0)$ to $(P_0/2, 3V_0)$
3. Straight line return path from $(P_0/2, 3V_0)$ back to $(P_0, V_0)$.
What is the net work done by the gas in one cycle?

2 moles of an ideal gas undergo an isothermal expansion at 400 K from volume $V$ to volume $e^3 V$. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
Calculate the change in entropy $\Delta S$ of the gas.

During an adiabatic process involving 1 mole of gas ($\gamma = 1.4$), the state changes from $(P_1 = 4.0 \times 10^5 \text{ Pa}, V_1 = 1.0 \times 10^{-3} m^3)$ to $(P_2 = 1.0 \times 10^5 \text{ Pa}, V_2 = 2.5 \times 10^{-3} m^3)$.
What is the work done by the gas during this process?

A monoatomic ideal gas ($\gamma = 5/3$) expands isobarically when heat is added to it.
What fraction of the total heat energy supplied is converted into external mechanical work?

A Carnot engine operating with a thermal efficiency of 25% produces 500 J of mechanical work per cycle.
How much heat is absorbed from the hot reservoir and exhausted to the cold reservoir each cycle?

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Grade 11 : Physics - Thermodynamics (Set 3)

Grade 11 : Physics - Thermodynamics (Set 3)— Questions & Detailed Solutions

A constant-volume gas thermometer measures a pressure of $1.20 \times 10^5 \text{ Pa}$ at the triple point of water (273.16 K).What temperature corresponds to a measured pressure of $1.50 \times 10^5 \text{ Pa}$ with this thermometer?

A sample of ideal gas undergoes an isobaric expansion at a constant pressure of $2.0 \times 10^5 \text{ Pa}$ from an initial volume of $0.010 m^3$ to a final volume of $0.025 m^3$.If 9000 J of heat is supplied to the gas during this process, what is the change in internal energy $\Delta U$?

Two moles of an ideal gas maintained at a constant temperature of 300 K expand isothermally from $V_1 = 2 \text{ L}$ to $V_2 = 2e^2 \text{ L}$ (where $e \approx 2.718$). Use $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.Calculate the work done by the gas during this isothermal expansion.

A monoatomic ideal gas ($\gamma = 5/3$) initially at 300 K is compressed adiabatically until its volume is reduced to $\frac{1}{8}$ of its initial volume.What is the final temperature of the gas?

Cyclic process $A \to B \to C \to A$ on a P-V diagram:State A: $(1 m^3, 100 \text{ kPa})$State B: $(4 m^3, 100 \text{ kPa})$State C: $(4 m^3, 400 \text{ kPa})$Path $C \to A$ is a straight line.What is the net work done by the gas during one complete cycle?

For a gas with ratio of heat capacities $\gamma = 1.40$, the universal gas constant is $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.What is the molar heat capacity at constant volume $C_v$?

1 mole of a rigid diatomic ideal gas ($C_v = \frac{5}{2}R$) is heated at constant volume from 300 K to 500 K. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.How much heat $Q$ is transferred to the gas?

Consider 2 moles of a non-linear triatomic ideal gas with 6 degrees of freedom ($f = 6$) at a temperature of 400 K. ($R = 8.31 \text{ J/(mol}\cdot\text{K)}$).Calculate the total internal energy $U$ of this gas sample.

A Carnot engine operates between a high-temperature reservoir at $227^\circ\text{C}$ and a low-temperature reservoir at $27^\circ\text{C}$.What is the maximum theoretical efficiency $\eta$ of this engine?

A refrigerator operates on a reversed Carnot cycle between $-23^\circ\text{C}$ and $27^\circ\text{C}$.What is the coefficient of performance (COP) of this refrigerator?

A heat engine absorbs 2000 J of thermal energy from a high-temperature source during each cycle and performs 600 J of mechanical work.Find the heat rejected to the cold reservoir and the thermal efficiency.

An ideal gas expands freely into an evacuated insulated chamber.Which set of thermodynamic parameters correctly describes this free expansion process?

A monoatomic ideal gas ($C_v = \frac{3}{2}R$) undergoes a polytropic process following $P V^2 = \text{constant}$.What is the molar heat capacity $C$ of the gas during this process?

Consider an isothermal process and an adiabatic process passing through the same point $(P_0, V_0)$ on a P-V diagram for an ideal gas with ratio of heat capacities $\gamma$.What is the ratio of the absolute slope of the adiabatic curve to that of the isothermal curve at this point?

A gas mixture consists of 1 mole of Helium ($C_v = \frac{3}{2}R$) and 1 mole of Oxygen ($C_v = \frac{5}{2}R$).What is the effective molar heat capacity at constant volume $C_{v,\text{mix}}$ of this mixture?

Cyclic process for an ideal gas:1. Isobaric expansion from $(P_0, V_0)$ to $(P_0, 3V_0)$2. Isochoric pressure drop from $(P_0, 3V_0)$ to $(P_0/2, 3V_0)$3. Straight line return path from $(P_0/2, 3V_0)$ back to $(P_0, V_0)$.What is the net work done by the gas in one cycle?

2 moles of an ideal gas undergo an isothermal expansion at 400 K from volume $V$ to volume $e^3 V$. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.Calculate the change in entropy $\Delta S$ of the gas.

During an adiabatic process involving 1 mole of gas ($\gamma = 1.4$), the state changes from $(P_1 = 4.0 \times 10^5 \text{ Pa}, V_1 = 1.0 \times 10^{-3} m^3)$ to $(P_2 = 1.0 \times 10^5 \text{ Pa}, V_2 = 2.5 \times 10^{-3} m^3)$.What is the work done by the gas during this process?

A monoatomic ideal gas ($\gamma = 5/3$) expands isobarically when heat is added to it.What fraction of the total heat energy supplied is converted into external mechanical work?

A Carnot engine operating with a thermal efficiency of 25% produces 500 J of mechanical work per cycle.How much heat is absorbed from the hot reservoir and exhausted to the cold reservoir each cycle?

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A constant-volume gas thermometer measures a pressure of $1.20 \times 10^5 \text{ Pa}$ at the triple point of water (273.16 K).
What temperature corresponds to a measured pressure of $1.50 \times 10^5 \text{ Pa}$ with this thermometer?

A sample of ideal gas undergoes an isobaric expansion at a constant pressure of $2.0 \times 10^5 \text{ Pa}$ from an initial volume of $0.010 m^3$ to a final volume of $0.025 m^3$.
If 9000 J of heat is supplied to the gas during this process, what is the change in internal energy $\Delta U$?

Two moles of an ideal gas maintained at a constant temperature of 300 K expand isothermally from $V_1 = 2 \text{ L}$ to $V_2 = 2e^2 \text{ L}$ (where $e \approx 2.718$). Use $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
Calculate the work done by the gas during this isothermal expansion.

A monoatomic ideal gas ($\gamma = 5/3$) initially at 300 K is compressed adiabatically until its volume is reduced to $\frac{1}{8}$ of its initial volume.
What is the final temperature of the gas?

Cyclic process $A \to B \to C \to A$ on a P-V diagram:
State A: $(1 m^3, 100 \text{ kPa})$
State B: $(4 m^3, 100 \text{ kPa})$
State C: $(4 m^3, 400 \text{ kPa})$
Path $C \to A$ is a straight line.
What is the net work done by the gas during one complete cycle?

For a gas with ratio of heat capacities $\gamma = 1.40$, the universal gas constant is $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
What is the molar heat capacity at constant volume $C_v$?

1 mole of a rigid diatomic ideal gas ($C_v = \frac{5}{2}R$) is heated at constant volume from 300 K to 500 K. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
How much heat $Q$ is transferred to the gas?

Consider 2 moles of a non-linear triatomic ideal gas with 6 degrees of freedom ($f = 6$) at a temperature of 400 K. ($R = 8.31 \text{ J/(mol}\cdot\text{K)}$).
Calculate the total internal energy $U$ of this gas sample.

A Carnot engine operates between a high-temperature reservoir at $227^\circ\text{C}$ and a low-temperature reservoir at $27^\circ\text{C}$.
What is the maximum theoretical efficiency $\eta$ of this engine?

A refrigerator operates on a reversed Carnot cycle between $-23^\circ\text{C}$ and $27^\circ\text{C}$.
What is the coefficient of performance (COP) of this refrigerator?

A heat engine absorbs 2000 J of thermal energy from a high-temperature source during each cycle and performs 600 J of mechanical work.
Find the heat rejected to the cold reservoir and the thermal efficiency.

An ideal gas expands freely into an evacuated insulated chamber.
Which set of thermodynamic parameters correctly describes this free expansion process?

A monoatomic ideal gas ($C_v = \frac{3}{2}R$) undergoes a polytropic process following $P V^2 = \text{constant}$.
What is the molar heat capacity $C$ of the gas during this process?

Consider an isothermal process and an adiabatic process passing through the same point $(P_0, V_0)$ on a P-V diagram for an ideal gas with ratio of heat capacities $\gamma$.
What is the ratio of the absolute slope of the adiabatic curve to that of the isothermal curve at this point?

A gas mixture consists of 1 mole of Helium ($C_v = \frac{3}{2}R$) and 1 mole of Oxygen ($C_v = \frac{5}{2}R$).
What is the effective molar heat capacity at constant volume $C_{v,\text{mix}}$ of this mixture?

Cyclic process for an ideal gas:
1. Isobaric expansion from $(P_0, V_0)$ to $(P_0, 3V_0)$
2. Isochoric pressure drop from $(P_0, 3V_0)$ to $(P_0/2, 3V_0)$
3. Straight line return path from $(P_0/2, 3V_0)$ back to $(P_0, V_0)$.
What is the net work done by the gas in one cycle?

2 moles of an ideal gas undergo an isothermal expansion at 400 K from volume $V$ to volume $e^3 V$. Take $R = 8.31 \text{ J/(mol}\cdot\text{K)}$.
Calculate the change in entropy $\Delta S$ of the gas.

During an adiabatic process involving 1 mole of gas ($\gamma = 1.4$), the state changes from $(P_1 = 4.0 \times 10^5 \text{ Pa}, V_1 = 1.0 \times 10^{-3} m^3)$ to $(P_2 = 1.0 \times 10^5 \text{ Pa}, V_2 = 2.5 \times 10^{-3} m^3)$.
What is the work done by the gas during this process?

A monoatomic ideal gas ($\gamma = 5/3$) expands isobarically when heat is added to it.
What fraction of the total heat energy supplied is converted into external mechanical work?

A Carnot engine operating with a thermal efficiency of 25% produces 500 J of mechanical work per cycle.
How much heat is absorbed from the hot reservoir and exhausted to the cold reservoir each cycle?