If specific heat of a system varies linearly as a function of temperature , does rate of heat absorbed also vary linearly with temperature?

Question

this is a question which was asked in JEE advanced 2013 . This is a multiple choice question. I am confident about options b) , c) and d) being right but I have a doubt in the correctness of option a).

As you may observe the graph is nearly linear in the region 0-100K , hence we may say that C(heat capacity) can be represented as' mT '(where m is slope and T is temperature ). We know that dQ=CdT (by definition of heat). Hence when we substitute for C we get dQ= mT dT , on integration we get a second degree term in T i.e option a) would be incorrect since it says Q varies linearly with T.

In all the solution/answers I have found for this problem they all mark option a) to be correct with the reasoning

Please tell me what mistake I am making in my reasoning or where I am going wrong .

Answer 1

You have misinterpreted the question. It does not say:

Q varies linearly with T

it says:

the rate at which heat is absorbed ...varies linearly with the temperature T

In your working you get as far as:

$$ \frac{dQ}{dT} = mT $$

We want the rate at which heat is absorbed, $dQ/dt$:

$$ \frac{dQ}{dt} = \frac{dQ}{dT}\frac{dT}{dt} = mT\frac{dT}{dt} \tag{1} $$

We are told that the temperature is increasing with time at a constant rate, so:

$$ T = at $$

for some constant $a$ and therefore:

$$ \frac{dT}{dt} = a $$

Substitute this in (1) and we get:

$$ \frac{dQ}{dt} = am\,T $$

i.e. the rate of heat absorption is linearly dependent on $T$.