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$: 0 K = -273°C.]$

Step 1: Identify the given parameters and convert temperatures to Kelvin.

Convert the temperatures from Celsius to Kelvin using the relation $T(\text{K}) = T(^\circ\text{C}) + 273$.

$$m = 1\ \text{kg}$$ $$T_1 = -73^\circ\text{C} = 200\ \text{K}$$ $$T_2 = 27^\circ\text{C} = 300\ \text{K}$$ $$C = kT$$ $$Q = nk$$

Step 2: Set up the differential equation for heat.

Since the specific heat capacity is temperature-dependent, express the heat required for an infinitesimal temperature change $dT$.

$$dQ = m C dT$$ $$dQ = m k T dT$$

Step 3: Integrate to find the total heat.

Integrate the expression from the initial temperature $T_1$ to the final temperature $T_2$.

$$Q = \int_{T_1}^{T_2} m k T \, dT$$ $$Q = mk \left[ \frac{T^2}{2} \right]_{T_1}^{T_2}$$ $$Q = \frac{mk}{2} \left( T_2^2 - T_1^2 \right)$$

Step 4: Substitute the values into the integrated equation.

Substitute the given values of mass and temperatures.

$$Q = \frac{1 \cdot k}{2} \left( 300^2 - 200^2 \right)$$ $$Q = \frac{k}{2} (90000 - 40000)$$ $$Q = 25000k$$

Step 5: Compare with the given heat expression to find $n$.

Equate the calculated heat to the given expression $Q = nk$.

$$nk = 25000k$$ $$n = 25000$$

Answer: 25000