Change in Internal Energy of a Heated sphere - Two Questions

If you heat a body, its temperature rises and so its internal energy increases. You have come across the increase in internal energy of a gas on many occasions and have noted the difference between the amounts of heat to be supplied on heating the gas at constant volume and at constant pressure to undergo a given temperature rise. Here are two questions involving the work done by a solid on heating it:

(1) If a solid of volume V is heated at constant pressure so as to have a small temperature rise, the work done by the solid will be directly proportional to

(a) V^1/2 (b) V^–1/2 (c) V (d) V^–1 (e) V⁰

(2) The temperature of a copper sphere of volume ‘V’m³ and density ‘ρ’ kg m^–3 increases by a small value ∆T when it absorbs a small quantity ∆Q joule of heat at atmospheric pressure ‘P’ pascal. If the specific heat of copper is ‘C’ J kg^–1K^–1, and the cubical expansivity of copper is ‘γ’ K^–1, the increase in internal energy of the copper sphere is

(a) ∆Q–PV

(b) ∆Q

(c) ∆Q(1– γPV/ρC)

(d) ∆Q(1– γP/ρC)

(e) ∆Q(1– γP/VρC)

Let us consider the second question first since it will give us the answer for the first question also.

The entire heat energy absorbed by the sphere is not used in increasing its internal energy. A small portion is used to do work in expanding (against the atmospheric pressure). Normally we omit this small amount of energy since the expansion of a solid is small. But you have to calculate it here since you are given all data for calculating it.

The rise in temperature of the sphere ∆T = ∆Q/mC where ‘m’ is the mass of the sphere.

Increase in volume of the sphere ∆V = Vγ∆T = (m/ρ)γ(∆Q/mC) = γ∆Q/ρC. [The density ρ can be assumed to be constant as the temperature rise is small].

Therefore, work done against the atmospheric pressure = P∆V = Pγ∆Q/ρC.

The increase in internal energy of the sphere = ∆Q – (Pγ∆Q/ρC) = ∆Q(1– γP/ρC).

So, the correct option is (d).

Now the answer for question No.1 is option (e) since the above expression for increase in internal energy does not contain the volume V.