91Ó°ÊÓ

Work done by the gas can be determined by the expression for work in a piston-cylinder assembly: \ \[ W = P \times A \times d \] \ Where: \ - \( P \) is the pressure exerted by the gas, including the pressure due to the weight of the piston and atmospheric pressure \ - \( A \) is the piston's face area \ - \( d \) is the elevation change of the piston \ \ First, calculate the total pressure: \ \[ P = P_{atm} + \frac{W_{piston} }{ A } \] \ \[ P = 14.7 \frac{\mathrm{lbf }}{ \mathrm{in}^{2} } + \frac{100 \mathrm{lbf }}{ 40 \mathrm{in}^{2} } = 14.7 \frac{ \mathrm{lbf} }{ \mathrm{in}^{2} } + 2.5 \frac{ \mathrm{lbf} }{\mathrm{in}^{2} } = 17.2 \frac{ \mathrm{lbf} }{ \mathrm{in}^{2} } \] \ Convert the face area (in inches squared) to feet squared: \ \[ A = 40 \frac { \mathrm{in}^{2} }{ (144 \mathrm{in}^{2} / \mathrm{ft}^{2}) } = 0.2778 \mathrm{ft}^{2} \] \ Now, the work done by the gas is: \ \[ W = P \times A \times d \] \ \[ W = 17.2 \frac{ \mathrm{lbf} }{ \mathrm{in}^{2} } \times 0.2778 \mathrm{ft}^{2} \times 1 \mathrm{ft} ( \frac{ 144 \mathrm{in}^{2} }{ \mathrm{ft}^{2} } ) = 688.9 \mathrm{lbf} \mathrm{ft} / 778 \mathrm{(Btu) } = 0.886 \mathrm{Btu} \]

The paddle wheel transfers 3 Btu of energy to the gas. \ Thus, \( Q = 3 \mathrm{Btu} \).

The first law of thermodynamics for a closed system without mass flow is: \ \[ \Delta U = Q - W \] \ Where: \ - \( \Delta U \) is the change in internal energy \ - \( Q \) is the heat added to the system (positive if added to the system) \ - \( W \) is the work done by the system (positive if done by the system) \ \ Plug in the values from the previous steps: \ \[ \Delta U = 3 \mathrm{Btu} - 0.886 \mathrm{Btu} = 2.114 \mathrm{Btu} \]