Consider a piston- cylinder arrangement with spring in its natural length. Due to heat transfer, the gas expands until the piston hits the stop. The mass of piston is `10kg` and area is `78cm^(2)`. Initial and final specific internal energies are 214 and `337kJ//kg`

Options

A. Initial pressure of gas is `112k` Pa
B. Work done by gas on the piston is `56.5J`
C. Work done by gas on the piston is `100J`
D. Heat transfer for 0.5 gas is `118J`

Explanation

To solve the problem involving a piston-cylinder arrangement with a spring, we need to analyze the given data and apply the principles of thermodynamics.

### Given Data:
- Mass of the piston, \( m = 10 \, \text{kg} \)
- Area of the piston, \( A = 78 \, \text{cm}^2 = 78 \times 10^{-4} \, \text{m}^2 = 0.0078 \, \text{m}^2 \)
- Initial specific internal energy, \( u_1 = 214 \, \text{kJ/kg} = 214000 \, \text{J/kg} \)
- Final specific internal energy, \( u_2 = 337 \, \text{kJ/kg} = 337000 \, \text{J/kg} \)

### Step 1: Calculate the change in internal energy per unit mass
The change in specific internal energy \( \Delta u \) can be calculated as:
\[ \Delta u = u_2 - u_1 = 337000 \, \text{J/kg} - 214000 \, \text{J/kg} = 123000 \, \text{J/kg} \]

### Step 2: Calculate the total change in internal energy for the piston
To find the total change in internal energy for the piston, we need to multiply the change in specific internal energy by the mass of the gas inside the cylinder. However, we need to find the mass of the gas first.

Assuming the gas is ideal and using the ideal gas law, we can express the mass of the gas in terms of its specific internal energy and the volume it occupies. For this problem, we will focus on the energy change.

### Step 3: Calculate the work done by the gas
The work done \( W \) by the gas during the expansion can be calculated using the formula:
\[ W = P \Delta V \]
Where \( P \) is the pressure and \( \Delta V \) is the change in volume. The pressure can be calculated using the force exerted by the piston divided by the area of the piston.

### Step 4: Calculate the force exerted by the piston
The force \( F \) exerted by the piston due to its weight is given by:
\[ F = m \cdot g = 10 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 98.1 \, \text{N} \]

### Step 5: Calculate the pressure exerted by the piston
The pressure \( P \) exerted by the piston can be calculated as:
\[ P = \frac{F}{A} = \frac{98.1 \, \text{N}}{0.0078 \, \text{m}^2} \approx 12500 \, \text{Pa} \]

### Step 6: Calculate the work done by the gas
Assuming the volume change is significant, we can express \( \Delta V \) in terms of the area and the distance moved by the piston. However, since the piston hits the stop, we can consider the work done as:
\[ W = P \cdot V \]
Where \( V \) can be approximated based on the area and the stroke length of the piston.

### Step 7: Apply the first law of thermodynamics
According to the first law of thermodynamics:
\[ \Delta U = Q - W \]
Where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added, and \( W \) is the work done by the system.

### Step 8: Solve for heat added
Rearranging the equation gives:
\[ Q = \Delta U + W \]

### Conclusion
To find the exact values for \( Q \) and \( W \), we would need more information about the volume change or the specific heat capacities of the gas involved. However, the steps outlined provide a framework for solving the problem based on the principles of thermodynamics.