# Derive heat capacity at constant pressure

• tjlaxs
Van der Waals equation with differentials.In summary, the problem is asking to derive the heat capacity at constant pressure (C_P) from the van der Waals equation for temperatures T >> T_C (critical temperature). To do this, the critical temperature is first derived from the fact that it exists at the point where (\frac{\mathrm{d}P}{\mathrm{d}V})_T = 0. However, this is not needed for the derivation. The correct equation for heat capacity (C_P) is Cp = dh/dt, where h is enthalpy, and to find h, the Van der Waals equation is used with differentials. One approach to
tjlaxs

## Homework Statement

For temperatures $$T >> T_C$$ (critical temperature) derive the heat capacity at constant pressure $$C_P$$ from van der Waals equation.

## Homework Equations

Critical temperature:
$$T_C = \frac{2N(V - Nb)^2}{kV^2}$$

$$T_C$$ is derived from the fact that it exist at the point in which
$$(\frac{\mathrm{d}P}{\mathrm{d}V})_T = 0$$ but I'm pretty certain that this is not needed in this derivation.

Van der Waals equation:
$$(P + aN^2/V^2)(V - Nb) = NkT$$

Heat capacity:
$$C_P = (\frac{\mathrm{d}U}{\mathrm{d}T})_P + P(\frac{\mathrm{d}V}{\mathrm{d}T})_P$$

## The Attempt at a Solution

I've tried to get the point in this. The first term in the equation of $$C_P$$ is easy, but the problem is the second term.

If I try to solve for the $$V$$ in the van der Waals equation I get a long equation set to derive. And I don't think this is what is the point of the exercise.

Is there another approach or something to simplify the van der Waals equation before the derivation?

It looks like you get a cubic equation in V to solve, which I agree is probably not what you're expected to do.

Just taking an educated guess here, but it's probably the case that
V >> Nb​
and
P >> aN2/V2
.
It may be a reasonable approximation to replace the "V2" term with whatever the ideal gas equation gives for V, since it appears as part of a term that is small to begin with.

You're using the wrong equation for hear capacity, you should use Cp = dh/dt

where h is enthaply

## 1. What is the definition of heat capacity at constant pressure?

Heat capacity at constant pressure is a measure of the amount of heat energy that is required to raise the temperature of a substance by one degree Celsius when the pressure on the substance remains constant.

## 2. How does heat capacity at constant pressure differ from heat capacity at constant volume?

Heat capacity at constant pressure takes into account the work done by the substance against the external pressure, while heat capacity at constant volume does not consider this work. This means that heat capacity at constant pressure is typically higher than heat capacity at constant volume.

## 3. What is the equation for calculating heat capacity at constant pressure?

The equation for heat capacity at constant pressure is Cp = q / ΔT, where Cp is the heat capacity at constant pressure, q is the heat added to the substance, and ΔT is the change in temperature.

## 4. How do you measure heat capacity at constant pressure in a laboratory setting?

To measure heat capacity at constant pressure, a calorimeter is used. The substance is placed in the calorimeter and heated while the pressure is kept constant. The change in temperature is recorded and used to calculate the heat capacity at constant pressure using the above equation.

## 5. What factors can affect the heat capacity at constant pressure of a substance?

The heat capacity at constant pressure of a substance can be affected by factors such as the molecular structure of the substance, the temperature, and any phase changes that occur during the heating process. It may also vary depending on the pressure and temperature range being considered.

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