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1. Molar Specific Heat Capacity at constant Volume ----- ##C_v##

2. Molar Specific Heat Capacity at constant Pressure ---- ##C_p##

And in case of Constant temperature there is no point in introducing Specific Heat Capacities.

##Doubt 1: ## So, we have Molar Specific Heat Capacities for cases when a gas is at a constant pressure or volume. But what if both the 'Volume' and 'Pressure' are varying?

Moreover my text says that, if a certain amount of energy is transferred to a gas the internal energy of a gas is same for any sort of process (isochoric, isobaric, isothermal, adiabatic). And it often uses the term ##nC_v\Delta T## for internal energy.

According to the first law of thermodynamics ##\Delta U = Q-W## (##W## - Work done 'by' the system).

So for an 'isochoric' process, as ##\Delta V = 0##, Work done is also zero (##W=0##).

Therefore: ##\Delta U=Q-0=nC_v\Delta T##

And in case of an 'isobaric' process it can be proved that ##\Delta U=Q-W=nC_p\Delta T - nR\Delta T=n\Delta T (C_p - R)=nC_v\Delta T##

But let's consider an adiabatic process. In the derivation of ##PV^\gamma=constant## the text gives the following relation:

##nC_v\Delta T = \Delta U = Q-W=0-W=-pdV## and proceeds.

##Doubt 2: ## But I don't understand the relation ##\Delta U = nC_v\Delta T##. In an adiabatic process the only condition is that no heat is transferred ##from## or ##to## the system. Both the 'volume' and 'pressure' and even the 'temperature' can be varying.

If I consider a gas in an adiabatic container, and if the gas expands, it does a positive work at the cost of the internal energy. This results in the reduction of the temperature of the gas. But how can the reduction of the temperature be related as ##\Delta U = nC_v\Delta T##?