Calculating Heat Quantity with Variable Specific Heat
The quantity of heat (\(Q\)) required to change the temperature of a substance is given by the integral of its mass (\(m\)) times its specific heat (\(C\)) over the temperature change (\(dT\)). When the specific heat is temperature-dependent, the formula becomes:
\( Q = \int_{T_1}^{T_2} m C(T) \, dT \)Given the specific heat \(C(T) = C_0 + \alpha T\), where \(C_0\) and \(\alpha\) are constants, we substitute this into the integral:
\( Q = m \int_{T_1}^{T_2} (C_0 + \alpha T) \, dT \)Integrating Specific Heat Function
We perform the integration with respect to temperature (\(T\)) from the initial temperature (\(T_1\)) to the final temperature (\(T_2\)):
\( Q = m \left[ C_0 T + \frac{\alpha T^2}{2} \right]_{T_1}^{T_2} \)Applying Temperature Limits
Now, we evaluate the integrated expression at the upper limit (\(T_2\)) and subtract the value at the lower limit (\(T_1\)):
\( Q = m \left( \left( C_0 T_2 + \frac{\alpha T_2^2}{2} \right) - \left( C_0 T_1 + \frac{\alpha T_1^2}{2} \right) \right) \)Group the terms involving \(C_0\) and \(\alpha\):
\( Q = m \left( C_0(T_2 - T_1) + \frac{\alpha}{2}(T_2^2 - T_1^2) \right) \)Simplifying the Expression
Using the difference of squares factorization (\(T_2^2 - T_1^2 = (T_2 - T_1)(T_1 + T_2)\)), we get:
\( Q = m \left( C_0(T_2 - T_1) + \frac{\alpha}{2}(T_2 - T_1)(T_1 + T_2) \right) \)Factor out the common term \(m(T_2 - T_1)\):
\( Q = m(T_2 - T_1) \left[ C_0 + \frac{\alpha}{2}(T_1 + T_2) \right] \)This matches the expression in Option D.
Similar Questions
- What is the mass of a material, whose specific heat capacity is 400 J/(kg °C) for a rise in temperature from 15°C to 25°C, when heat received is 20 kJ ?
Important Questions from Calorimetry
Which of the following statements is INCORRECT for heat?
A copper block of mass 3 kg is heated in a furnace to a temperature of 450° C and then placed on a large ice block. Find the maximum amount of ice that can melt? (specific heat of copper = 0.39 Jg-1K-1, heat of fusion of water = 335 Jg-1K-1)
- When steam at 100°C is passed into 60 g of water at 10°C, the temperature of water rises to 40°C. What will be the total mass of water (in g) at 40°C?
- What is the mass of a material, whose specific heat capacity is 400 J/(kg °C) for a rise in temperature from 15°C to 25°C, when heat received is 20 kJ ?
