# Adiabatic process-how is internal energy mcv(t2-t1)

• sreeram
In summary, in an ideal gas, the internal energy change (ΔU) is equal to mass (m) multiplied by the constant volume specific heat (Cv) multiplied by the change in temperature (T2-T1). This equation applies to any process, not just constant volume processes. This is because for ideal gases, the internal energy is only dependent on temperature, not pressure or volume. This is due to the assumption that there is no interaction between gas particles. This relationship was derived from the equation dU=TdS-pdV and the assumption that pressure is proportional to temperature at constant volume.
sreeram
In polytropic process internal energy change is=mCv(T2-T1)? why is it not mCp(T2-T1)? Why are we not taking specific heat at constant pressure?

Last edited:
Are you dealing with an ideal gas? If so, the internal energy is $U=mC_VT+U_0=m(C_P-R)T+U_0$ (the internal energy depends only on temperature), so $\Delta U=mC_V\Delta T=m(C_P-R)\Delta T$. $C_V$ (or $C_P-R$) are just constants here. The equations apply to any process.

mCv(t2-t1) is the heat required to raise the temperature of mass 'm' from t1 to t2 at constant volume. But adiabatic process is not a constant volume process. My question is how does this equation give the value of change in internal energy?

Are you dealing with an ideal gas?

yes. I am dealing with ideal gas

It's a unique property of ideal gases that $\Delta U=mC_V\Delta T$ holds for any process. The constant-volume constraint is not required. That's why I also wrote the equation as $\Delta U=m(C_P-R)\Delta T$, which similarly doesn't mean that constant pressure is required just because the constant-pressure specific heat appears. Again, $C_V$ and $C_P$ are just acting as constants, and do not represent constraints on the equation. It's hard for every thermo student to get used to this, but that's how it is.

Thanks for the answer. But how did they found this relation?

It comes from one of the assumptions we make for an ideal gas, that there's no interaction between the atoms or molecules. If this is true, then changing the pressure (while keeping temperature constant) shouldn't have any effect on the total energy (i.e., $(\partial U/\partial P)_T=0$). We're left with a energy dependence on temperature alone.

Starting from
$$\mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V$$
one can in fact show that for the special case
$$p=Tf(V)$$
i.e. pressure proportional to temperature at constant volume, the energy is a function of temperature only $U=A(T)$.

The proportionality to temperature doesn't follow from it yet, but if you also assume that the density of states (statmech treatment) is a power law, the you can show that $U=CT$.

## 1. What is an adiabatic process?

An adiabatic process is a thermodynamic process in which there is no exchange of heat between a system and its surroundings. This means that the internal energy of the system remains constant.

## 2. How is internal energy related to adiabatic processes?

Internal energy is the total energy contained within a system, including both its kinetic and potential energy. In an adiabatic process, the internal energy remains constant because there is no change in heat, only work is done on or by the system.

## 3. What is the formula for calculating internal energy in an adiabatic process?

The formula for calculating internal energy in an adiabatic process is U = mcV(T2 - T1), where U is the internal energy, m is the mass of the system, c is the specific heat capacity, V is the volume, and T2 - T1 is the change in temperature.

## 4. How does the specific heat capacity affect the internal energy in an adiabatic process?

The specific heat capacity is a measure of how much energy is required to raise the temperature of a substance. In an adiabatic process, the internal energy is directly proportional to the specific heat capacity. This means that a higher specific heat capacity will result in a higher internal energy for the same change in temperature.

## 5. What is the significance of the change in temperature in an adiabatic process?

The change in temperature in an adiabatic process is important because it is directly related to the change in internal energy. A larger change in temperature will result in a larger change in internal energy, and vice versa. This is because the temperature is a measure of the average kinetic energy of the particles in the system, and a change in internal energy is a measure of the overall energy of the system.

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