Solutions for Exercises of Thermodynamics

1. Choose the correct option.

i) A gas in a closed container is heated with 10J of energy, causing the lid of the container to rise 2m with 3N of force. What is the total change in energy of the system?

(A) 1 OJ

(B) 4J

(D) – 4J

ii) Which of the following is an example of the first law of thermodynamics?

(A) The specific heat of an object explains how easily it changes temperatures.

(B) While melting, an ice cube remains at the same temperature.

(D) After falling down the hill, a ball’s kinetic energy plus heat energy equals the initial potential energy.

iii) Efficiency of a Carnot engine is large when

(A) T_H is large

(B) T_c is low

(D) T_H – T_c is small

iv) The second law of thermodynamics deals with transfer of:

(A) work done

(B) energy

(D) heat

v) During refrigeration cycle, heat is rejected by the refrigerant in the :

(A) condenser

(B) cold chamber

(D) hot chamber

2. Answer in brief.

i) A gas contained in a cylinder surrounded by a thick layer of insulating material is quickly compressed. (a) Has there been a transfer of heat? (b) Has work been done?

ii) Give an example of some familiar process in which no heat is added to or removed form a system, but the temperature of the system changes.

iii) Give an example of some familiar process in which heat is added to an object, without changing its temperature.

iv) What sets the limits on efficiency of a heat engine?

v) Why should a Carnot cycle have two isothermal two adiabatic processes?

3. Answer the questions.

i) A mixture of hydrogen and oxygen is enclosed in a rigid insulting cylinder. It is ignited by a spark. The temperature and the pressure both increase considerably. Assume that the energy supplied by the spark is negligible, what conclusions may be drawn by application of the first law of thermodynamics?

ii) A resistor held in running water carries electric current. Treat the resistor as the system (a) Does heat flow into the resistor? (b) Is there a flow of heat into the water? (c) Is any work done? (d) Assuming the state of resistance to remain unchanged, apply the first law of thermodynamics to this process.

iii) A mixture of fuel and oxygen is burned in a constant-volume chamber surrounded by a water bath. It was noticed that the temperature of water is increased during the process. Treating the mixture of fuel and oxygen as the system, (a) Has heat been transferred ? (b) Has work been done? (c) What is the sign of AU ?

iv) Draw a p-V diagram and explain the concept of positive and negative work. Give one example each.

v) A solar cooker and a pressure cooker both are used to cook food. Treating them as thermodynamic systems, discuss the similarities and differences between them.

4. A gas contained in a cylinder fitted with a frictionless piston expands against a constant external pressure of 1 atm from a volume of 5 litres to a volume of 10 litres. In doing so it absorbs 400 J of thermal energy from its surroundings. Determine the change in internal energy of system.

P = 1 atm = 1.013 × 10⁵N/m²,

V = 5 lit = 5 × 10^-3m³,

V = 10 lit = 10^-2m³,

Q = 400 J, ΔU = ?

According to first law of thermodynamics, we have,

$latex Q=\triangle U+PdV$

$latex \therefore\triangle U=Q-PdV=\;Q-P(V_2-V_1)$

$latex \therefore\triangle U=400-1.013\times10^5\lbrack10^{-2}-5\times10^{-3}\rbrack$

$latex \therefore\triangle U=400-1.013\times10^5\times10^{-2}\lbrack1-0.5\rbrack$

$latex \therefore\;\triangle U\;=\;400\;-\;506.5$

$latex \therefore\;\triangle U\;=\;-106.5J$

$latex \therefore\;\left|\triangle U\right|\;=\;106.5J$

5. A system releases 125 kJ of heat while 104 kJ of work is done on the system. Calculate the change in internal energy.

Q = −125kJ = −125 × 10³J,

W = −104kJ = −104 × 10³J,

ΔU = ?

According to first law of thermodynamics, we have,

$latex Q=\triangle U+W$

$latex \therefore\triangle U=Q-W$

$latex =(125\times10^3)-(-104\times10^3)$

$latex\triangle U\;=\;(-125+104)\times10^3$

$latex \therefore\;\left|\triangle U\right|\;=\;21kJ$

6. Efficiency of a Carnot cycle is 75%. If temperature of the hot reservoir is 727°C, calculate the temperature of the cold reservoir.

$latex \eta\;=\;75\%\;=\;\frac{75}{100}\;=\;0.75,$

$latex T_1\;=\;T_H\;=\;727^\circ C\;=\;1000K$

$latex T_2=T_c=?$

We know that efficiency of Carnot cycle is given as,

$latex \eta=1-\frac{T_2}{T_1}$

$latex \therefore\;\frac{T_2}{T_1}\;=\;1-\eta$

$latex \therefore\;\frac{T_2}{T_1}\;=\;1-0.75\;=\;0.25$

$latex \therefore\;T_2\;=\;0.25\times1000\;=\;250K$

$latex \therefore\;T_{2\;}=\;-23^\circ C$

7. A Carnot refrigerator operates between 250°K and 300°K. Calculate its coefficient of performance.

T = T = 250K, T = T = 300K , K = ?

We know that Coefficient of performance of Carnot refrigerator is given as,

$latex K=\frac{T_2}{T_1-T_2}$

$latex \therefore\;K\;=\;\frac{250}{300-250}$

$latex \therefore\;K\;=\;\frac{250}{50}\;=\;5$

8. An ideal gas is taken through an isothermal process. If it does 2000 J of work on its environment, how much heat is added to it?

As the process is isothermal, internal energy remains constant. Hence heat supplied is completely converted into the work.

∴ Heat added = Work done

i.e Work done = 2000J

9. An ideal monatomic gas is adiabatically compressed so that its final temperature is twice its initial temperature. What is the ratio of the final pressure to its initial pressure?

$latex T_f\;=\;2T_i,\;\frac{P_f}{P_i}\;=\;?$

For ideal monoatomic gas

$latex we\;have,\;\gamma=\frac53$

For adiabatic process,

$latex we\;have,\;PV^\gamma=cons\tan t(c)$

$latex \therefore\;P=\left(\frac{nRT}P\right)^\gamma=c\;\;\;\;\;\;\;as\;PV=nRT$

$latex \therefore P^{1-\gamma}T^\gamma=\frac c{{(nR)}^\gamma}=cons\tan t$

$latex \therefore\;P_f^{1-\gamma}T_f^\gamma\;=\;P_i^{1-\gamma}T_i^\gamma$

$latex \therefore\frac{P_f^{1-\gamma}}{P_i^{1-\gamma}}=\frac{T_i^\gamma\;\;}{T_f^\gamma}$

$latex \therefore\;\left(\frac{P_f}{P_i}\right)^{1-\gamma}\;=\;\left(\frac{T_i}{T_f}\right)^\gamma$

$latex \therefore\;\left(\frac{P_f}{P_i}\right)^{1-\frac53}\;=\;\frac12^\frac53$

$latex \therefore\;\left(\frac{P_f}{P_i}\right)^{-\frac23\times-\frac32}$

$latex =\;\left(\frac12\right)^{\frac53\times-\frac32}$

$latex \therefore\;\frac{P_f}{P_i}\;=\;\left(\frac12\right)^{-\frac52}$

$latex \therefore\;\frac{P_f}{P_i}\;=\;2^{5/2}$

$latex \therefore\;\frac{P_f}{P_i}\;=\;\sqrt{2^5}\;=\;\sqrt{32}$

$latex \therefore\;\frac{P_f}{P_i}\;=\;5.65$

10. A hypothetical thermodynamic cycle is shown in the figure. Calculate the work done in 25 cycles.

We know that work done in a thermodynamic process is area under the P-V curve. But given process is cyclic process with path as a complete ellipse. Hence work done in this process will be area of the ellipse.

$latex \therefore W=\pi ab$

Here,a = semi − major axis,b = semi − minor axis

$latex\therefore\;W\;=\;\pi\times\frac{(6-2)\times10^{-3}}2\times\frac{(11-1)\times10^5}2$

$latex \therefore\;W\;=\;\pi\times10^3$

$latex \therefore W=3.14\times10^3\;Joule$

11. The figure shows the V-T diagram for one cycle of a hypothetical heat engine which uses the ideal gas. Draw (a) the p–V diagram of the system.
[Ans: (a)] [Ans: (b)]

12. A system is taken to its final state from initial state in hypothetical paths as shown figure calculate the work done in each case.

Work done along AB

W = PΔV

W = 6 × 10⁵ × (6 − 2)

W = 24 × 10⁵

∴ W_AB = 2.4 × 10⁶J

Work done along BC is zero as ΔV = 0.

Work done along CD

W = PΔV

W = 2 × 10⁵ × (6 − 2)

W = 8 × 10⁵

∴ W_CD= 8 × 10⁵J

Work done along DA is zero as ΔV = 0.