Derive the expression for the work done by the gas during isothermal expansion.

We know work done by or on the system is given by

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Derive the equation of state for adiabatic process. Plot P versus V graph for the process.

According to the first law of thermodynamics,

In adiabatic process, no heat is allowed to exchange between the system and surrounding, therefore dQ = 0. Thus,

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An ideal gas equation is,

PV = nRT

Since in isothermal process, T is constant, therefore the equation of state for isothermal process reduces to

PV = constant

P versus V graph and P versus T graph for isothermal process is as shown below.

According to first law of thermodynamics,

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Derive an expression for the work done by a gas undergoing expansion from volume V_{1} to V_{2}.

Let us consider an ideal gas enclosed in a cylinder fitted with massless and frictionless piston. Let A be the area of cross-section of piston. Let V be the volume and P be the pressure exerted by gas on the piston. The piston is kept in equilibrium by applying pressure P from outside.

Let the applied pressure be decreased by infinitesimally small amount, so that the piston moves by infinitesimal distance dx. The amount of work done by the gas in infinitesimal expansion is,

where dV = Adx, the infinitesimal increase in volume. Total work done by gas in expanding it from volume V

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Derive an expression for the work done in adiabatic process.

Consider a unit mole of gas contained in a perfectly non-conducting cylinder provided with a non-conducting and frictionless piston.

Let C

Let at any instant, when the pressure of gas is P, the gas be compressed by small volume dV.

Then work done on the gas is,

dW = PdV

Total work done on gas to compress from volume V

...(1)

According to first law of thermodynamics,

For adiabatic process,

∴ PdV = -dU =

∴

where,

T

Thus, work done is given by,

Therefore,

The above expression gives us the amount of work done in adiabatic process.

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