Confusion in calculating $\Delta U$ from a bomb calorimeter

Question

In the book, it is mentioned the formula for $\Delta U$ in a bomb calorimeter without any derivation:

$$\Delta U = q_v = \frac{Q\times M\times \Delta T}{m}$$ where $$Q=\textrm{heat capacity of calorimeter,}$$ $$M=\textrm{molecular mass of sample,}$$ $$m=\textrm{mass of sample used, and}$$ $$\Delta T=\textrm{change in temperature of water in the bath}$$

I am confused regarding this formula. Can anyone give me the derivation of this formula (or a corrected formula)?

[I am 11-grader and am studying chemical thermodynamics. I can distinguish between $C$ as an extensive property and $c$ and $C_m$ as intensive properties.]

Any help would be appreciated :)

NOTE: I know that a formula is $q_v=cm\Delta T$, I want to know how the book got to the formula mentioned previously.

Answer 1

The formula in the book is correct. They are trying to get the change in internal energy per mole of sample. From the first law, for this constant volume system (no work), $$\Delta U_{\textrm{total}}=q=C\Delta T$$where C is the heat capacity of the calorimeter. This equation assumes that the heat capacity of the water in the bath is lumped into C, and that the temperature change of other parts of the calorimeter is the same as that of the water.

The number of moles of sample is m/M. So, $$\Delta U_{\textrm{per mole}}=\Delta U_{\textrm{total}}\frac{M}{m}=C\Delta T\frac{M}{m}$$In their notation, they use the symbol Q to represent the heat capacity of the calorimeter C.