Show that for an ideal gas CP - CV = R

1.	Show that for an ideal gas CP - CV = R
Answer» When a gas is heated under constant pressure conditions, the heat is required for raising the temperature of the gas and also for doing mechanical work against the external pressure during expansion. Consider one mole of gas to expand from volume V₁ to V₂ at a constant pressure p, when the temperature is raised from T K to (T + 1) K. Then by definition C_P - C_V = work done by the gas due to expansion. Work done by the gas during expansion = pΔV = p(V₂ - V₁) Therefore, C_P - C_V = p(V₂ - V₁) For one mole of the gas, using gas equation, one can write pV₂ = R (T + 1) and pV₁ = RT or, pV₂ - pV₁ = R(T + 1) - RT = R(T + 1) - RT = R or, p(V₂ - V₁) = R Therefore, C_P - C_V = R

Answer»

When a gas is heated under constant pressure conditions, the heat is required for raising the temperature of the gas and also for doing mechanical work against the external pressure during expansion.

Consider one mole of gas to expand from volume V₁ to V₂ at a constant pressure p, when the temperature is raised from T K to (T + 1) K. Then by definition

C_P - C_V = work done by the gas due to expansion. Work done by the gas during expansion

= pΔV = p(V₂ - V₁)

Therefore, C_P - C_V = p(V₂ - V₁)

For one mole of the gas, using gas equation, one can write

pV₂ = R (T + 1)

and pV₁ = RT

or, pV₂ - pV₁ = R(T + 1) - RT

= R(T + 1) - RT = R

or, p(V₂ - V₁) = R

Therefore, C_P - C_V = R

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