An insulated piston-cylinder device initially contains 300 L of air at 120 kPa a

To solve the problem, we need to calculate the entropy change of air under two assumptions: (a) constant specific heats and (b) fluctuating specific heats.
(a) Constant Specific Heats:
The entropy change ($\Delta S$) can be calculated using the formula:
$\Delta S={\int }_{T_1}^{T_2}\frac{C_p}{T}dT-R\ln \left(\frac{V_2}{V_1}\right)$
where $T_1$ and $T_2$ are the initial and final temperatures, $C_p$ is the heat capacity at constant pressure, $V_1$ and $V_2$ are the initial and final volumes, and $R$ is the specific gas constant.
Given:
Initial air pressure ($P_1$) = 120 kPa
Initial volume ($V_1$) = 300 L
Initial temperature ($T_1$) = 17 °C = 290 K
Power supplied ($P$) = 200 W
Heating time ($t$) = 15 minutes = 900 s
We can find the final temperature ($T_2$) using the formula:
$P_1·V_1=P_2·V_2$
Substituting the values, we can solve for $P_2$:
$120\times 300=P_2\times V_2$
Now, we can calculate the final temperature using the ideal gas law:
$P_2·V_2=n·R·T_2$
where $n$ is the number of moles of air.
Assuming air as an ideal gas, we can find $n$ using the ideal gas equation:
$P_1·V_1=n·R·T_1$
Once we have $n$, we can solve for $T_2$.
Next, we can calculate the entropy change using the given formula.
(b) Fluctuating Specific Heats:
The entropy change ($\Delta S$) can be calculated using the formula:
$\Delta S={\int }_{T_1}^{T_2}\frac{C_p(T)}{T}dT-R\ln \left(\frac{V_2}{V_1}\right)$
In this case, $C_p(T)$ is the heat capacity at constant pressure as a function of temperature.
We can use the heat capacity at constant pressure for air ($C_p$) as a function of temperature, which can be approximated as:
$C_p(T)=a+bT+c{T}^2+d{T}^3$
where $a$, $b$, $c$, and $d$ are constants.
Substituting this expression into the entropy change formula, we can calculate the entropy change using numerical integration.
Once we have both results, we can compare the entropy changes under the two assumptions.
Calculations:
(a) Constant Specific Heats:
1. Calculate the final temperature $T_2$:
$P_1·V_1=P_2·V_2$
2. Calculate the number of moles $n$:
$n=\frac{P_1·V_1}{R·T_1}$
3. Calculate $T_2$ using the ideal gas law:
$P_2·V_2=n·R·T_2$
4. Calculate the entropy change using the formula:
$\Delta S={\int }_{T_1}^{T_2}\frac{C_p}{T}dT-R\ln \left(\frac{V_2}{V_1}\right)$
(b) Fluctuating Specific Heats:
1. Define the constants $a$, $b$, $c$, and $d$ for the heat capacity expression.
2. Calculate the entropy change using numerical integration:
$\Delta S={\int }_{T_1}^{T_2}\frac{C_p(T)}{T}dT-R\ln \left(\frac{V_2}{V_1}\right)$
Finally, compare the two entropy change results.
Using these calculations, we can find the entropy change of air with the given parameters.

Step 1: Find the final temperature of the air using the constant specific heats assumption.
Given:
Initial pressure ($P_1$) = 120 kPa
Initial volume ($V_1$) = 300 L
Initial temperature ($T_1$) = 17 °C
Final volume ($V_2$) = 300 L (as the piston-cylinder apparatus is insulated)
Heat transfer ($Q$) = 200 W
Time ($t$) = 15 minutes
To find the final temperature ($T_2$) using the constant specific heats assumption, we can use the ideal gas law:
$P_1{V}_1/T_1=P_2{V}_2/T_2$
Substituting the given values:
$(120\times {10}^3\text{Pa})·(300\text{L})/(17+273\text{K})=P_2·(300\text{L})/T_2$
Simplifying:
$120·300/(17+273)=P_2/T_2$
Step 2: Calculate the change in entropy using the constant specific heats assumption.
Entropy change ($\Delta S$) can be calculated using the equation:
$\Delta S=C_p\ln (T_2/T_1)-R\ln (V_2/V_1)$
where $C_p$ is the specific heat capacity at constant pressure and $R$ is the specific gas constant.
For air, $C_p=1.005\text{kJ/kg K}$ and $R=0.287\text{kJ/kg K}$.
Substituting the values and solving the equation, we can find $\Delta S$.
Step 3: Calculate the change in entropy using the fluctuating specific heats assumption.
When specific heats vary with temperature, we can calculate the entropy change using the equation:
$\Delta S={\int }_{T_1}^{T_2}\frac{C_p}{T}dT-R\ln (V_2/V_1)$
where $C_p$ is the specific heat capacity at constant pressure and $R$ is the specific gas constant.
Integrating the equation and evaluating it between the initial and final temperatures will give us $\Delta S$.
Note: To perform the integration and obtain a closed-form solution, we need the specific heat capacity as a function of temperature. If the equation for $C_p(T)$ is provided, we can use it to calculate the integral.