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

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.

• Does this answer your question? Calculating the heat capacity of a calorimeter Commented Jan 1, 2021 at 7:24
• No, I know that formula and want to know how the book got to the aforementioned formula @cngzz1! Commented Jan 1, 2021 at 7:25
• I took a very quick look at this, it might helps you: chem.tamu.edu/class/fyp/labquest/inv16.html Commented Jan 1, 2021 at 8:41
• Hi the previous link was the derivation for constant pressure specific heat capacity. Here's a derivation for constant volume specific heat capacity ( which explains your calorimeter experiment): chemistry.stackexchange.com/q/49198/102629 Commented Jan 1, 2021 at 12:48
• @ultralegend5385 number of mole is m/M, right. Multiply X time M/m gives the wanted quantity per mole (1/mol). . Commented Jan 1, 2021 at 16:34

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.