Final answer:
To calculate the temperature of the gas, we need to first determine the work done on the gas. Using the ideal gas law and the equation for work, we can solve for the temperature. The temperature of the gas is approximately 1500 K.
Step-by-step explanation:
In this question, we are given that it takes 500 J of work to compress quasi-statically 0.50 mol of an ideal gas to one-fifth its original volume. We are asked to calculate the temperature of the gas, assuming it remains constant during the compression.
First, let's determine the work done on the gas using the equation for work:
Work (W) = Pressure (P) * Change in Volume (ΔV)
Since the gas is compressed and the volume decreases to one-fifth its original volume, the change in volume, ΔV, is -4/5 times the original volume. Therefore, ΔV = -4/5 * V.
Plugging in the values, we have:
500 J = P * (-4/5 * V)
Now, we need to express the pressure in terms of moles and temperature using the ideal gas law:
PV = nRT
Since the temperature remains constant, we can solve P from the equation:
P = (nRT) / V
Substituting the value of P into the equation for work, we get:
500 J = ((nRT) / V) * (-4/5 * V)
Simplifying, we have:
500 J = -4/5 * nRT
Now, we can solve for the temperature, T:
T = (500 J * -5) / (-4 * n * R)
Where R is the ideal gas constant. Since we are given the number of moles, we can substitute the value into the equation and calculate the temperature:
T = (500 J * -5) / (-4 * 0.50 mol * 8.314 J/(mol*K))
Solving this equation gives us a temperature of approximately 1500 K.