Let's say we have [math]m[/math] kg of oxygen at [math]T_1[/math] K. After heating the oxygen, it did [math]A[/math] J of work. Find the temperature to which the oxygen was heated. [math]p=const[/math]. This is all simple of course, using the [math]A=p(V_2-V_1)=\frac{m}{M}R(T_2-T_1)[/math] formula. Now we are able to find how much did [math]U[/math] change using the [math]U=\frac{3m}{2M}R(T_2-T_1)[/math] formula. Since [math]Q=U+A[/math] according to the first law of thermodynamics, [math]Q=\frac{5m}{2M}R(T_2-T_1)[/math]. Here's the problem:

[math]Q=cm(T_2-T_1)=\frac{5m}{2M}R(T_2-T_1)[/math]

from witch

[math]c=\frac{5R}{2M}[/math]. Numbers don't show that the given formula is correct nor is it correct when we take [math]V=const, c=\frac{3R}{2M}[/math]. So, my question is: Why is the above all wrong, and how do you calculat c with ideal gases and how to make a small triangle appear in front of [math]T[/math]?

Oxygen is a diatomic molecule and hence has 6 degrees of freedom, 3 translational, 2 rotational and 1 vibrational. Both rotational degrees of freedom will contribute to Cv but the vibrational one won't (at least at normal temperatures).

For a diatomic Cv = 5/2R, Cp =7/2R