The document The First Law Of Thermodynamics Heat Capacity (Part - 2) - Heat, Irodov JEE Notes | EduRev is a part of the Course I. E. Irodov Solutions for Physics Class 11 & Class 12.

**Q. 44. Demonstrate that the process in which the work performed by an ideal gas is proportional to the corresponding increment of its internal energy is described by the equation pV ^{n} = const, where n is a constant. **

**Solution. 44. **According to the problem : A α U or dA = aU (where a is proportionality constant)

or, (1)

From ideal gas law, pV= v R T, on differentiating

pdV + Vdp = v RdT (2)

Thus from (1) and (2)

or,

Dividing both the sides by pV

On integrating n In V + In p = In C (where C is constant)

or,

**Q. 45. Find the molar heat capacity of an ideal gas in a polytropic process pV ^{n} = const if the adiabatic exponent of the gas is equal to γ. At what values of the polytropic constant n will the heat capacity of the gas be negative? **

**Solution. 45. **In the polytropic process work done by the gas

(where T_{i} and T_{f} are initial and final temperature of the gas like in adiabatic process)

and

By the first law of thermodynamics Q = ΔU + A

According to definition of molar heat capacity when number of moles v = 1 and ΔT = 1 then Q = Molar heat capacity.

Here,

**Q. 46. In a certain polytropic process the volume of argon was increased α = 4.0 times. Simultaneously, the pressure decreased β = 8.0 times. Find the molar heat capacity of argon in this process, assuming the gas to be ideal.**

**Solution. 46. **Let the process be polytropic according to the law pV^{n} = constant

Thus,

So,

In the polytropic process molar heat capacity is given by

So,

**Q. 47. One mole of argon is expanded polytropically, the polytropic constant being n = 1.50. In the process, the gas temperature changes by ΔT = — 26 K. Find: (a) the amount of heat obtained by the gas; (b) the work performed by the gas**

**Solution. 47. **(a) Increment of internal energy for ΔT, becomes

From first law of thermodynamics

(b) Sought work done,

**Q. 48. An ideal gas whose adiabatic exponent equals y is expanded according to the law p = αV , where a is a constant. The initial volume of the gas is equal to V _{0}. As a result of expansion the volume increases η times. Find: (a) the increment of the internal energy of the gas; (b) the work performed by the gas; (c) the molar heat capacity of the gas in the process.**

**Solution. 48. **LaW 0f the process is p = α V or pV^{-1} = α

so the process is polytropic of index n = - 1

As p = αV so, P_{i} - αV_{0 }and p_{f} = α η V_{0}

(a) Increment of the internal energy is given by

(b) Work done by the gas is given by

(c) Molar heat capacity is given by

**Q. 49. An ideal gas whose adiabatic exponent equals γ is expanded so that the amount of heat transferred to the gas is equal to the decrease of its internal energy. Find: (a) the molar heat capacity of the gas in this process; (b) the equation of the process in the variables T, V; (c) the work performed by one mole of the gas when its volume increases η times if the initial temperature of the gas is T**

**Solution. 49. **

where C_{n} is the molar heat capacity in the process. It is given that

So,

(b) By the first law of thermodynamics, dQ - dU + dA,

or,

So,

or,

(c)

But from part (a), we have

Thus

From part (b); we know

So, (where T is the final temperature)

Work done by the gas for one mole is given by

**Q. 50. One mole of an ideal gas whose adiabatic exponent equals y undergoes a process in which the gas pressure relates to the temperature as p = aT ^{α}, where a and α are constants. Find: (a) the work performed by the gas if its temperature gets an increment ΔT; (b) the molar heat capacity of the gas in this process; at what value of α will the heat capacity be negative? **

**Solution. 50. **Given p = a T^{α} (for one mole of gas)

So,

or,

Here polytropic exponent

(a) In the poly tropic process for one mole of gas :

(b) Molar heat capacity is given by

**Q. 51. An ideal gas with the adiabatic exponent γ undergoes a process in which its internal energy relates to the volume as U = aV ^{α}, where a and α are constants. Find: (a) the work performed by the gas and the amount of heat to be transferred to this gas to increase its internal energy by ΔU; (b) the molar heat capacity of the gas in this process. **

**Solution. 51.**

or,

or,

or

So polytropric index n = 1 - α

(a) Work done by the gas is given by

Hence

By the first law of thermodynamics, Q = ΔU+A

(b) Molar heat capacity is given by

**Q. 52. ****An ideal gas has a molar heat capacity C _{v} at constant volume. Find the molar heat capacity of this gas as a function of its volume V, if the gas undergoes the following process: **

**Solution. 52. **By the first law .of thermodynamics

Molar specific heat according to definition

We have

After differentiating, we get

So,

Hence

(b)

So,

**Q. 53. One mole of an ideal gas whose adiabatic exponent equals γ undergoes a process p = p _{0} + α /V, w here P_{0} and α are positive constants. Find:**

(a) heat capacity of the gas as a function of its volume;

(b) the internal energy increment of the gas, the work performed by it, and the amount of heat transferred to the gas, if its volume increased from V_{1} to V_{2}.

**Solution. 53. **Using 2.52

(for one mole of gas)

Therefore

Hence

(b) Work done is given by

By the first law of thermodynamics Q = ΔU +A

**Q. 54. ****One mole of an ideal gas with heat capacity at constant pressure C _{p} undergoes the process T = T_{0} + αV, where T_{0} and α are constants. Find: **

(a) heat capacity of the gas as a function of its volume;

(b) the amount of heat transferred to the gas, if its volume increased from V_{1} to V_{2}.

**Solution. 54. (a) **Heat capacity is given by

(see solution of 2.52)

We have

After differentiating, we get,

Hence

(b)

for one mole of gas

Now

By the first law of thermodynamics Q = ΔU + A

**Q. 55. For the case of an ideal gas find the equation of the process (in the variables T, V) in which the molar heat capacity varies as: (a) C = C _{v} + αT; (b) C = C_{v} + βV; (c) C = C_{v} + ap, where α, β, and a are constants. **

**Solution. 55. **Heat capacity is given by

Integrating both sides, we get is a constant.

Or,

and

or,

Integrating both sides, we get

So,

So,

or

or,

or V - a T = constant

**Q. 56. An ideal gas has an adiabatic exponent γ. In some process its molar heat capacity varies as C = α/T, where α is a constant. Find: (a) the work performed by one mole of the gas during its heating from the temperature T _{0} to the temperature η times higher; (b) the equation of the process in the variables p, V. **

**Solution. 56. **(a) By the first law of thermodynamics A = Q - ΔU

or,

Given

So,

(b)

Given

or,

or,

or,

Integrating both sides, we get

or,

or,

or,

or,

**Q. 57. Find the work performed by one mole of a Van der Waals gas during its isothermal expansion from the volume V _{1} to V_{2} at a temperature T. **

**Solution. 57. **The work done is

**Q. 58. One mole of oxygen is expanded from a volume V _{1} = 1.00 1 to V_{2} = 5.0 l at a constant temperature T = 280 K. Calculate:**

(a) the increment of the internal energy of the gas:

(b) the amount of the absorbed heat.

The gas is assumed to be a Van der Waals gas.

**Solution. 58. **(a) The increment in the internal energy is

But from second law

On the other hand

or,

So,

(b) From the first law

**Q. 59. For a Van der Waals gas find: (a) the equation of the adiabatic curve in the variables T, V; (b) the difference of the molar heat capacities C**

**Solution. 59. **(a) From the first law for an adiabatic

dQ = dU + pd V = 0

From the previous problem

So,

This equation can be integrated if we assume that Cv and b are constant then

or,

(b) We use

Now,

So along constant p,

Thus

On differentiating,

or,

and

**Q. 60. Two thermally insulated vessels are interconnected by a tube equipped with a valve. One vessel of volume V _{1} = 10 l contains v = 2.5 moles of carbon dioxide. The other vessel of volume V_{2} = 100 l is evacuated. The valve having been opened, the gas adiabatically expanded. Assuming the gas to obey the Van der Waals equation, find its temperature change accompanying the expansion. **

**Solution. 60. **From the first law

as the vessels are themally insulated.

As this is free expansion,

But

So,

or,

Substitution gives ΔT = - 3 K

**Q. 61. What amount of heat has to be transferred to v = 3.0 moles of carbon dioxide to keep its temperature constant while it expands into vacuum from the volume V _{1} = 5.0 l to V_{2} = 10 l ? The gas is assumed to be a Van der Waals gas. **

**Solution. 61. ** (as A = 0 in free expansion).

So at constant temperature.

= 0.33 kJ from the given data.

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