# Validly of the equation, du= nCv dt, for an adiabatic process [duplicate]

In an adiabatic process, there is no heat exchange, hence, $$Q=0$$.

The equation $$\mathrm dU=nC_V\,\mathrm dT$$ is valid for every type of process given the gas is ideal, where $$C_V$$ is heat at constant volume required to raise the temperature of a system by 1°, q. We know that $$Q=0$$ for an adiabatic process, So $$\mathrm dU$$ = Work(adiabatic) And we define, $$W$$ (for an adiabatic process)$$=nC_V\,\mathrm dT$$ But how can $$C_V$$ have a numerical value for an adiabatic process, there is no heat exchange, no matter how much heat we provide, temperature of system would not increase, shouldn’t $$C_V=0$$? But then the equation won’t be valid.

Is the equation , $$\mathrm dU=nC_V\,\mathrm dT$$ even valid for an adiabatic process?

I don’t get it, where am I wrong? Please help me out, it’s my first time on here, apologies for any error.

• Where do you think the gas takes the energy to do the work? There is no gas elastic energy as if you release a compressed rubber. It is taken for kinetic energy of molecules = inner thermal energy, what means T lowering so W = n.C_v.Delta T. (both sides negative) May 8, 2023 at 18:18
• In thermodynamics, $C_V$ is defined in terms of U, not in terms of q. May 8, 2023 at 18:41
• May 8, 2023 at 19:27

Consider a closed system for which $$\mathrm{d}n=0$$.
Then, the definition of $$C_v$$ is given by $$\left(\dfrac{\partial U}{\partial T}\right)_V$$.
$$C_v$$ is an intrinsic property of any material and it doesn't depend on the process involved.
If heat was provided at constant volume, it would raise the temperature of the system, and $$C_v$$ is a measure of this phenomenon.
In an adiabatic process, the work is still equal to $$\int_{T_1}^{T_2}C_v\mathrm{d}T$$. The temperature is changing in adiabatic processes because of the work and not the heat, which can both change the temperature of the system.