A quantity of gas occupies a volume of 0.3 m^3 at a pressure of 101kPa and a temperature of 19°C.The gas is compressed isothermally to a pressure of 500 kPa then expanded adiabatically to its initial volume. Assume gamma =1.4 and R= 0.288kj/kg.K . Determine the heat recieved or rejected
To determine the heat received or rejected in this process, we can use the First Law of Thermodynamics, which states that the change in internal energy (∆U) of a system is equal to the heat added to the system (Q) minus the work done by the system (W):
∆U = Q - W
In this case, the process involves two steps: an isothermal compression and an adiabatic expansion. Let's calculate the heat received or rejected for each step individually.
Step 1: Isothermal Compression
During the isothermal compression, the temperature remains constant (T1 = T2 = 19°C = 292K), so there is no change in internal energy (∆U = 0). Therefore, the equation simplifies to:
0 = Q1 - W1
We need to calculate the work done by the system during this step. The work done in an isothermal process can be calculated using the formula:
W1 = nRT ln(V2/V1)
Where:
n = number of moles of the gas
R = specific gas constant (in this case, 0.288 kJ/kg·K)
T = temperature (in Kelvin)
V1 = initial volume
V2 = final volume
Given:
P1 = 101 kPa
P2 = 500 kPa
V1 = 0.3 m³
First, we need to determine the number of moles of the gas. We can use the ideal gas law equation:
PV = nRT
Rearranging the equation, we get:
n = PV / RT
n = (P1 * V1) / (R * T1)
n = (101 * 0.3) / (0.288 * 292)
Now we can calculate the work done during the isothermal process:
W1 = nRT ln(V2/V1)
W1 = [(101 * 0.3) / (0.288 * 292)] * (0.288 * 292) * ln(V2/0.3)
Since the volume is compressed, V2 will be less than 0.3 m³. Let's assume V2 = 0.25 m³. Plug in these values into the equation and solve for W1.
W1 = [(101 * 0.3) / (0.288 * 292)] * (0.288 * 292) * ln(0.25/0.3)
Step 2: Adiabatic Expansion
During the adiabatic expansion, there is no heat exchanged with the surroundings (Q2 = 0). Therefore, the equation simplifies to:
∆U = -W2
We need to calculate the work done by the system during this step. The work done in an adiabatic process can be calculated using the formula:
W2 = C/(γ - 1) * (P1 * V1 - P2 * V2)
Where:
γ = specific heat ratio (in this case, 1.4)
P1 = initial pressure
P2 = final pressure
V1 = initial volume
V2 = final volume
Given:
P1 = 500 kPa
P2 = 101 kPa
V1 = 0.25 m³
Now we can calculate W2:
W2 = C/(γ - 1) * (P1 * V1 - P2 * V2)
W2 = (0.288 * 292)/(1.4 - 1) * (500 * 0.25 - 101 * 0.3)
Finally, to determine the total heat received or rejected, we sum the heat and work values during each step:
Q = Q1 + Q2 = W1 + (-W2)
Simplify the equation:
Q = W1 - W2
Plug in the calculated values for W1 and W2, and you will get the answer to the heat received or rejected in this process.