# Isothermal expansion of real gas - pressure change

I have been studying isothermal expansion process, the heat change particulary for a real gas. On the internet is many solved problems about this, but it's always about changing volume from $$v_i$$ to $$v_f$$. But what if I want to study how the presure changes? Because volume and pressure are tied, but for real gas I cannot use the law stating $$\;p_iv_i=$$ const. explicitly.

The state equations for one mole of the real gas are:

$$p = \frac {RT}{v-b} - \frac {a}{v^2}$$ $$\;$$ and $$\;$$ $$u = cRT - \frac{a}{v}$$, $$\;$$ where $$a$$ and $$b$$ are constants.

Then for the volume change we have:

$$\int_{v_i}^{v_f}\frac {RT}{v-b} - \frac {a}{v^2}dV = [RTln(v-b)+a/v]^{v_f}_{v_i}$$

• You seem to be assuming that the expansion is isothermal and reversible. Is that correct? – Chet Miller Jan 19 '19 at 17:06
• Yes, I do assume it. And I made the mistake with that interval energii part so I am going to edit it. – Andromeda Jan 19 '19 at 18:37
• My question is maybe broader,... what to do, if my differential form is expressed in the way I dont want it. I was thinking about expressing dV as total differential with the respect of the pressure and temperature. – Andromeda Jan 19 '19 at 18:38
• The mathematics is such that you can't do that analytically. The equation in cubic in v. Why don't you follow @Philip Wood's suggestion? – Chet Miller Jan 19 '19 at 19:03
• I want to calculate the heat change, I need to integrate. So do you mean calculate the heat via standard volume change then express initial and final volume from the state equation and plug it inside? Unfortunately its not possible exactly to express the volume in terms of pressure and temperature for real gas. – Andromeda Jan 19 '19 at 19:30

"For isothermal process we have that $$dQ=pdv,$$ because $$du$$ is zero in this case."
This is the case for an ideal gas, for which $$U$$ is a function of $$T$$ only, but it is not generally true for a real gas.
• I think the problem is that I misinterpreted the question. I thought you wanted to find the pressure change – in terms of $what?$ – Philip Wood Jan 19 '19 at 18:48
• So, just to be clear, you want to find the pressure change for a given heat input, $\Delta Q?$ Or do you want to find the heat input or output for a given change of pressure? Or neither?\$ – Philip Wood Jan 19 '19 at 19:23