The specific heat capacity of a substance is temperature dependent and is given by the formula 𝐶 = 𝑘𝑇, where 𝑘 is a constant of suitable dimensions in SI units, and 𝑇 is the absolute temperature. If the heat required to raise the temperature of 1 kg of the substance from −73 °C to 27 °C is 𝑛𝑘, the value of 𝑛 is _____.

[Given: 0 K = −273 °C.]

The correct answer is 25000.

Step-by-step Explanation:
We are given that the specific heat capacity $C$ of a substance depends on its absolute temperature $T$ according to the relation:

$C=kT$

where $k$ is a constant.
The mass of the substance is $m=1\text{kg}$.

First, let's convert the given temperatures from Celsius to Kelvin using the relation $T\left(\text{K}\right)=T\left({\text{°C}}^{}\right)+273$:
Initial temperature, $T_1=-73{\text{°C}}^{}=-73+273=200\text{K}$
Final temperature, $T_2=27{\text{°C}}^{}=27+273=300\text{K}$

The heat energy $Q$ required to raise the temperature of a mass $m$ when the specific heat capacity is temperature-dependent is given by the integral:

$Q={\int }_{T_1}^{T_2}mCdT$

Substitute the values $m=1$ and $C=kT$ into the equation:

$Q={\int }_{200}^{300}\left(1\right)\left(kT\right)dT$

$Q=k{\int }_{200}^{300}TdT$

Integrating $T$ with respect to $T$ gives:

$Q=k{\left[\frac{T^2}{2}\right]}_{200}^{300}$

$Q=\frac{k}{2}\left[{300}^2-{200}^2\right]$

$Q=\frac{k}{2}\left[90000-40000\right]$

$Q=\frac{k}{2}\left[50000\right]$

$Q=25000k$

We are given that the heat required is $nk$. Comparing this with our calculated heat $Q$:
$nk=25000k$
Therefore, $n=25000$.