Relation between First Law of Thermodynamics and Ideal Gas Law
Thermodynamics has always been a tough thing for me. There are lots of assumptions in this subject (those assumptions, I know, are necessary,I know the science of thermodynamics is a very practical science).
First Law of Thermodynamics states mathematically :-
$$\Delta U = q + W $$ (with proper sign conventions must be used). This is just a law of conservation of energy and a very straightforward equation, but when we come to chemical thermodynamics this equation changes it's form and becomes :- $$\Delta U = q + P\Delta V $$ My intuition says as soon as Pressure and Volume comes in any equation it becomes specifically for gases. So, my first question is :-
Why thermodynamical equations are just for gases?
Let's imagine an isothermal expansion of a gas (that simple piston and gas experiment) under a constant pressure, now work W is $$W = P\Delta V$$ but if use Ideal Gas Law equation i.e. $$ PV = nRT$$ $$P\Delta V = \Delta n RT + nR\Delta T ~~~~~~~~~~~(1) $$
since the expansion is isothermal therefore $\Delta T = 0$ and I can think that during expansion no atom or molecule has been annihilated therefore $\Delta n = 0 $, so after all we get $$P\Delta V = 0$$ $$ W=0$$
I want to know my mistakes in above consideration.
There is a question in my book :-
A swimmer coming out of a pool is covered with a film of water weighing 18g. 749
How much heat must be supplied to evaporate this water at 298K ? Calculate the internal energy change of vaporization at $100^0 C$. $\Delta _{vap}H^0$ for water at $373 K = 40.66 kJ mol^{-1}$
My book give its solution like this
$$ H_2O (l) \longrightarrow H_2O(g)$$
Number of moles 18g of $H_2O$ (l) is just 1 mole. Since, $\Delta U = q - P\Delta V $ , therefore,
$$ \Delta U = \Delta H - P\Delta V $$. $$ \Delta U = \Delta H - \Delta n RT$$ $$ \Delta U = 40.66 \times (10)^3 Jmol^{-1} - 1\times(8.314 JK^{-1}mol^{-1}\times373K$$ $$ \Delta U = 37.56 kJmol^{-1}$$.
I have lot of problems with this solution which goes directly to the foundations of science of thermodynamics. (I must say it's because of these books that science becomes a rotten subject, these books destroy the real essence of science).
How is $\Delta n = 1$ ?
Why temperature is taken as 373 K and not 298 K, since the process starts at 298 K we should use it?
At 373 K the process becomes an isothermal one (latent heat) so $\Delta U$ ought to be zero, if we think the process of vaporization starts from 373 K.
Any help will be much appreciated. Thank you. 565