Step 1: Understanding the Topic
This question asks to calculate the change in enthalpy ($\Delta H$) for a substance when it is heated. The heat capacity at constant pressure ($C_p$) is given as a function of temperature, which means we cannot use the simple formula $\Delta H = nC_p\Delta T$. Instead, we must use the integral form of the relationship between enthalpy and heat capacity.
Step 2: Key Formula or Approach
When $C_p$ is temperature-dependent, the change in enthalpy is calculated by integrating the $C_p$ function with respect to temperature over the specified range. The formula for one mole is:
\[
\Delta H = \int_{T_1}^{T_2} C_p \, dT
\]
First, we must convert the given temperatures from Celsius to Kelvin.
Step 3: Detailed Calculation
A. Convert Temperatures to Kelvin:
The initial and final temperatures are:
\[
T_1 = 27^\circ C + 273.15 = 300.15\,K \approx 300\,K
\]
\[
T_2 = 327^\circ C + 273.15 = 600.15\,K \approx 600\,K
\]
(Using 300 K and 600 K for simplicity as is common in such problems).
B. Set up and Solve the Integral:
Substitute the given expression for $C_p$ into the integral:
\[
\Delta H = \int_{300}^{600} (a + bT) \, dT
\]
Integrate the expression term by term:
\[
\Delta H = \left[ aT + \frac{bT^2}{2} \right]_{300}^{600}
\]
Now, evaluate the definite integral:
\[
\Delta H = \left( a(600) + \frac{b(600)^2}{2} \right) - \left( a(300) + \frac{b(300)^2}{2} \right)
\]
This can be simplified as:
\[
\Delta H = a(600 - 300) + \frac{b}{2}(600^2 - 300^2)
\]
C. Substitute the Numerical Values:
Given $a = 19.5$ and $b = 0.042$:
\[
\Delta H = 19.5(300) + \frac{0.042}{2}(360000 - 90000)
\]
\[
\Delta H = 5850 + 0.021(270000)
\]
\[
\Delta H = 5850 + 5670
\]
\[
\Delta H = 11520
\]
Step 4: Final Answer
The change in enthalpy is 11,520. The units would be Joules if $C_p$ is in J/K·mol.
\[
\boxed{\Delta H = 11520 \text{ J}}
\]