To determine which path has the greatest heat transferred to the gas and by what percent it is greater, we need to compare the heat transferred for both the isothermal and adiabatic paths.
Let's begin by calculating the heat transferred in the isothermal path.
In an isothermal process, the temperature remains constant. We can use the ideal gas law to calculate the initial and final pressures of the gas.
For the initial state:
PV = nRT
Given:
Volume (V1) = 2.0 L
R (Universal gas constant) = 8.314 J/(mol·K)
Temperature (T1) = constant
Number of moles (n) = constant
From the ideal gas law, we can rearrange the equation to solve for the initial pressure (P1) as:
P1 = (nRT1) / V1
Similarly, for the final state:
Volume (V2) = 3.0 L
Using the same equation, we can solve for the final pressure (P2) as:
P2 = (nRT1) / V2
Now, let's calculate the work done in the isothermal process using the formula:
W = nRT1 * ln(V2/V1)
Since the process is isothermal, the heat transferred (Q) is equal to the work done:
Q(isothermal) = W = nRT1 * ln(V2/V1)
Next, let's calculate the heat transferred in the adiabatic path.
In an adiabatic process, no heat is transferred to or from the gas. Therefore, the equation for heat transferred is simply:
Q(adiabatic) = 0
Now, let's compare the two heat transfers.
The heat transferred in the isothermal path is given by:
Q(isothermal) = nRT1 * ln(V2/V1)
The heat transferred in the adiabatic path is:
Q(adiabatic) = 0
Since Q(isothermal) is greater than Q(adiabatic) in this case, the heat transferred to the gas is greatest for the isothermal path.
To calculate the percentage by which it is greater, we can use the following formula:
Percentage increase = (Q(isothermal) - Q(adiabatic)) / Q(adiabatic) * 100
Substituting the values, we get:
Percentage increase = (Q(isothermal) - 0) / 0 * 100 = Infinity
Therefore, the heat transferred in the isothermal path is infinitely greater than that in the adiabatic path.