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Compute the specific heat capacities at constant volume and constant pressure for air at standard temperature and pressure, assuming it is diatomic ideal gas and a molecular mass of 28u.

I have the equation for an ideal gas : $pV=Nk_BT$

and the caloric equation is : $E=\frac{5}{2}Nk_BT$

Now my definition for heat capacity is: $C=\frac{\delta Q}{dT}$. (Specific heat capacity will this divided by the mass).

Now I get that

$$ C=\frac{\delta Q}{dT}=\frac{dE+pdV}{dT}$$

and

$$C_V=\left(\frac{\partial E}{\partial T} \right)_V \ \ \ \text{,} \ \ \ C_p=\left(\frac{\partial E}{\partial T} \right)_p+p\left(\frac{\partial V}{\partial T} \right)_p$$

I tried $C_V=\left(\frac{\partial E}{\partial T} \right)_V=\left(\frac{5}{2}Nk_B \right)_V$

but I cannot see how I can proceed with this. Nor do I think $C_p$ will be any easier to use

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  • $\begingroup$ don't you know $C_v=(f/2)R,C_p=C_v+R$ $\endgroup$ – RE60K Feb 10 '15 at 18:22
  • $\begingroup$ I cannot see that in my notes $\endgroup$ – Permian Feb 10 '15 at 18:23
  • $\begingroup$ haven't you studied thermodynamics under physics or chemistry? $\endgroup$ – RE60K Feb 10 '15 at 18:25
  • $\begingroup$ no just in maths $\endgroup$ – Permian Feb 10 '15 at 18:25
  • $\begingroup$ wierd connection: thermodynamics and maths....hmm $\endgroup$ – RE60K Feb 10 '15 at 18:25
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In physics it's $q=\Delta U+w$ and in chemistry it's $\Delta U=q+w$, don't know what it is in maths. but anyways I start with chemistry definition, since every such equation differs only by a sign.

At constant volume, $dV=0$, so $$\Delta W=-pdV=0$$
so $$\Delta Q=\Delta E=\left(\frac{5}{2}\right)Nk_B\Delta T$$
so $$C_v=\frac{\partial Q}{\partial T}=\left(\frac{5}{2}\right)Nk_B$$ At constant pressure, $$\Delta H=\Delta E+\Delta (pV)=\Delta Q+\Delta W+pdV=\Delta Q+(-pdV)+pdV=\Delta Q$$ so $$\Delta Q=\Delta H=\Delta E+\Delta(pV)=\left(\frac{5}{2}\right)Nk_B\Delta T+\underbrace{Nk_B\Delta T}_{\text{actually it's }\Delta(Nk_BT) }=\left(\frac{5}{2}+1\right)Nk_BT$$, so $$C_p=\frac{\partial Q}{\partial T}=\left(\frac{5}{2}+1\right)Nk_B$$

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  • $\begingroup$ $\Delta E = \Delta Q + \Delta W$, Q heat, W work done, E internal energy? $\endgroup$ – Permian Feb 10 '15 at 18:59
  • $\begingroup$ @1234 yes here, delta E is delta U, delta Q is q and delta W is w $\endgroup$ – RE60K Feb 10 '15 at 19:09
  • $\begingroup$ Why are you using deltas when this is a question about something stable? $\endgroup$ – Permian Feb 10 '15 at 19:20
  • $\begingroup$ @1234 if you give heat and are trying to characterise it using heat capacity... then a change is bound to happen $\endgroup$ – RE60K Feb 10 '15 at 19:22
  • $\begingroup$ When are we "giving" heat? $\endgroup$ – Permian Feb 10 '15 at 20:03

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