A vertical hollow cylinder contains an ideal gas. The gas is enclosed by a 5 kg movable piston with a cross section area of 5x10^-3 m^3. The gas is heated from 300 to 350K and piston rises by 0.1m. Piston is clamped and gas is cooled to 300K. Find difference between heat added during the heating process and that lost during the cooling process.

To find the difference between the heat added during the heating process and the heat lost during the cooling process of the gas in the cylinder, we can use the principles of thermodynamics, specifically focusing on the ideal gas law and the concepts of work done by the gas.

Understanding the Heating Process

When the gas is heated from 300 K to 350 K, it expands and does work on the piston. The work done by the gas can be calculated using the formula:

• Work (W) = Pressure (P) × Change in Volume (ΔV)

The change in volume can be determined from the cross-sectional area of the piston and the distance it rises:

• ΔV = Area (A) × Height (h)
• ΔV = 5 × 10^-3 m² × 0.1 m = 5 × 10^-4 m³

Calculating Pressure

To find the pressure, we can use the ideal gas law:

• P = nRT/V

However, we need to know the number of moles (n) of the gas. We can find it using the initial conditions. Assuming the gas behaves ideally, we can rearrange the ideal gas law to find n:

• n = PV/RT

For simplicity, let's assume the pressure remains constant during the heating process. The initial pressure can be calculated using the initial temperature (T1 = 300 K) and the volume (V = A × h_initial). If we assume the initial height of the gas is h_initial, we can express the initial volume as:

• V_initial = A × h_initial

Heat Added During Heating

The heat added (Q) during the heating process can be calculated using the formula:

• Q = nC_vΔT

Where C_v is the specific heat capacity at constant volume. For an ideal gas, we can use the molar specific heat capacity. Assuming air as an ideal gas, C_v is approximately 718 J/(kg·K). The change in temperature (ΔT) is:

• ΔT = T_final - T_initial = 350 K - 300 K = 50 K

Now, we can calculate the heat added:

• Q_heating = n × C_v × ΔT

Cooling Process

When the gas is cooled back to 300 K, the piston is clamped, meaning no work is done during this process. The heat lost (Q_lost) can be calculated similarly:

• Q_lost = nC_vΔT

Here, ΔT for the cooling process is:

• ΔT = T_initial - T_final = 300 K - 300 K = 0 K

Since there is no change in temperature during cooling, the heat lost is zero:

• Q_lost = 0

Calculating the Difference

The difference between the heat added during the heating process and the heat lost during the cooling process is:

• Difference = Q_heating - Q_lost

Since Q_lost is zero, the difference is simply equal to the heat added during the heating process:

• Difference = Q_heating

In conclusion, the heat added during the heating process is the total difference since no heat is lost during the cooling process. This demonstrates how energy transfer works in thermodynamic systems, especially with ideal gases and movable pistons.