Six moles of an ideal gas are in a cylinder fitted at one end with a movable piston. The initial temperature of the gas is 27.0°C and the pressure is constant. As part of a machine design project, calculate the final temperature of the gas after it has done 2.40 * 10^3 J of work.

Verified step by step guidance

1. Convert the initial temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature. This is because the calculations in thermodynamics are typically done in Kelvin.
2. Use the first law of thermodynamics, which states that the change in internal energy (\(\Delta U\)) of a system is equal to the heat added to the system (Q) minus the work done by the system (W). In this case, the formula simplifies to \(\Delta U = -W\) because the pressure is constant and no heat is added or removed.
3. Calculate the change in internal energy (\(\Delta U\)) using the work done by the gas. Since the work is done by the gas, it will be negative when calculating \(\Delta U\).
4. Relate the change in internal energy to the change in temperature using the equation \(\Delta U = nC_V\Delta T\), where \(n\) is the number of moles, \(C_V\) is the molar heat capacity at constant volume, and \(\Delta T\) is the change in temperature. Solve for \(\Delta T\).
5. Add the change in temperature (\(\Delta T\)) to the initial temperature (in Kelvin) to find the final temperature of the gas in Kelvin. Convert this final temperature back to Celsius if required by subtracting 273.15.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. This law is fundamental in understanding the behavior of gases under various conditions, particularly when analyzing changes in temperature and volume while keeping pressure constant.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. In the context of the gas doing work, this principle implies that the work done by the gas will result in a change in internal energy, which is related to temperature changes in the gas.

Work Done by a Gas

When a gas expands against a piston, it does work on the piston, which can be quantified as W = PΔV, where P is pressure and ΔV is the change in volume. In this scenario, since the pressure is constant, the work done by the gas will directly affect its internal energy and, consequently, its temperature.