An ideal gas whose adiabatic exponent equals γ , is expanded according to the law p = α V , where, α 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.

Given, the aw of the process is p = α V or p V - 1 = α , where, α is constant. So, the process is polytropic of index n = - 1 . As p = α V , so, p i = α V 0 and p f = α η V 0 . ( a ) Change in internal energy, ∆ U = v C V ∆ T So, the increment of the internal energy is given by, Δ U = v R γ - 1 T f - T i = p f V f - p i V i γ - 1 ∴ Δ U = α V 0 2 η 2 - 1 γ - 1 ( b ) Work done by the gas in polytropic process is given by, A = p i V i - p f V f n - 1 A = α V 0 2 - α η V 0 · η V 0 - 1 - 1 A = α V 0 2 1 - η 2 - 2 ∴ A = 1 2 α V 0 2 η 2 - 1 ( c ) The molar heat capacity in a polytropic process is C n = R γ - 1 - R n - 1 C n = R ( n - γ ) ( n - 1 ) ( γ - 1 ) C n = R ( - 1 - γ ) ( - 1 - 1 ) ( γ - 1 ) ∴ C n = R 2 γ + 1 γ - 1 .

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

Important Questions on THERMODYNAMICS AND MOLECULAR PHYSICS

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.48.

An ideal gas whose adiabatic exponent equals

γ

, is expanded according to the law

p = α V

, where,

α

is a constant. The initial volume of the gas is equal to

C_{n} = \frac{8.314}{1.66 - 1} - \frac{8.314 \ln 4}{\ln 8 - \ln 4} = - 42 J {mol}^{- 1} K^{- 1}

. 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.

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.51.

An ideal gas with the adiabatic exponent

γ

, undergoes a process in which its internal energy relates to the volume as

U = a V^{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

Δ U

(b)

the molar heat capacity of the gas in this process.

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.52.

An ideal gas has a molar heat capacity

C_{V}

at a constant volume. Find the molar heat capacity of this gas as a function of its volume

V

, if the gas undergoes the following process,

(a)

T = T_{0} e^{α V}

(b)

p = p_{0} e^{α V}

,
where,

T_{0}, p_{0}

and

α

, are constants.

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.53.

One mole of an ideal gas whose adiabatic exponent equals

γ

, undergoes a process

p = p_{0} + \frac{α}{V}

, where

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 is increased from

V_{1}

V_{2}

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.54.

One mole of an ideal gas with heat capacity at constant pressure

C_{p}

, undergoes the process

T = T_{0} + α V

, 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 is increased from

V_{1}

V_{2}

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.55.

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} + α T

(b)

C = C_{V} + β V

(c)

C = C_{V} + a p

, where

α, β

and

a

Are constants.

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.56.

An ideal gas has an adiabatic exponent

γ

. In some process, its molar heat capacity varies as

C = \frac{α}{T}

, where

α

is a constant. Find

(a)

The work performed by one mole of the gas, during its heating from temperature

T_{0}

to a temperature,

η

times higher.

(b)

The equation of the process, in the variables

p, V

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JEE Main

IMPORTANT

Problems in General Physics>Chapter 2 - THERMODYNAMICS AND MOLECULAR PHYSICS>THE FIRST LAW OF THERMODYNAMICS, HEAT CAPACITY>Q 2.57.

Find the work performed by one mole of a Van der Waals gas, during its isothermal expansion from volume

V_{1}

V_{2}

, at a temperature

T

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

Important Questions on THERMODYNAMICS AND MOLECULAR PHYSICS

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