Final answer:
The specific work done by the highly cooled compressor can be calculated using the formula: Work = Pressure change x Volume of gas x Natural logarithm of final pressure divided by initial pressure. By substituting the given values into the formula, the specific work done can be obtained.
Step-by-step explanation:
In an isothermal process, the temperature remains constant. Therefore, the specific work done by the highly cooled compressor can be calculated using the formula:
Work = Pressure change x Volume of gas x Natural logarithm of final pressure divided by initial pressure
Given that the initial pressure is 100 kPa and the final pressure is 800 kPa, and using the ideal gas law, we can calculate the volume of the gas before and after compression to find the pressure change.
By substituting the values into the formula, the specific work done by the compressor can be obtained.
To calculate the specific work done during the isothermal compression of hydrogen gas, one can use the equation w = RT × ln(P2/P1) and plug in the given values for temperature and pressures.
Finding the Specific Work in an Isothermal Compression Process
The student has asked about the specific work involved in an isothermal compression process of hydrogen gas from an initial state of 300 K and 100 kPa to a final pressure of 800 kPa. To find the specific work assuming a reversible process, we use the formula for work done during an isothermal process for an ideal gas:
W = nRT × ln(P2/P1)
where:
n is the number of moles of the gas,
R is the universal gas constant,
T is the absolute temperature of the gas,
P1 is the initial pressure,
P2 is the final pressure.
As specific work is the work done per mole, we can simplify the equation to:
w = RT × ln(P2/P1)
By inserting the given values (T = 300 K, P1 = 100 kPa, and P2 = 800 kPa) into the formula, we can calculate the specific work done by the gas during isothermal compression:
w = (8.314 J/mol·K)(300 K) × ln(800/100)
After calculating the natural logarithm and multiplying by RT, we get the specific work.