Exemplar Problems

Problem 8 : The heat capacity of a substance is given by C = 5T, where C is the heat capacity in J/K and T is the temperature in K. Calculate the change in internal energy (∆U) when the temperature of 2 moles of this substance is raised from 273 K to 373 K.

Solution :

The change in internal energy (∆U) for a substance with a heat capacity C that varies with temperature is given by the equation:

∆U = ∫C dT

In this case, C = 5T, so we can integrate:

∆U = ∫(5T) dT from 273 K to 373 K

∆U = 5 ∫T dT from 273 K to 373 K

∆U =

5 [T²/2] from 273 K to 373 K

∆U = 5 [(373²/2) - (273²/2)]

∆U ≈ 5 [139,129.5 J - 74,529.5 J] ≈ 5 [64,600 J] ≈ 323,000 J ≈ 323 kJ

So, the change in internal energy (∆U) when the temperature is raised from 273 K to 373 K is approximately 323 kJ.