An ideal gas has the following initial conditions: Vi = 490 cm3, Pi = 5 atm, and Ti = 100°C. What is its final temperature if the pressure is reduced to 1 atm and the volume expands to 1000 cm3?
To determine the final temperature of the gas, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
To solve for the final temperature, we need to know the number of moles of the gas. However, since the initial and final states of the gas are not given, we cannot directly calculate the moles.
In this case, we can assume that the gas is an ideal gas, which means that it follows the ideal gas law throughout the process, and thus the number of moles remains constant. This assumption is valid if we have a gas that is not close to undergoing a phase change.
Since we don't know the number of moles directly, we can use the initial conditions (Vi = 490 cm3, Pi = 5 atm, and Ti = 100°C) to find the initial number of moles (ni) using the ideal gas law equation:
PiVi = niRTi
To solve for ni, we can rearrange the equation:
ni = (PiVi) / (RTi)
Now we can use the final conditions (Pi = 1 atm, Vf = 1000 cm3) and the initial number of moles (ni) to find the final temperature (Tf). Rearranging the ideal gas law equation:
(TfPiVf) = (niR)
Solving for Tf:
Tf = (niR) / (PiVf)
Substituting the expressions for ni and R:
Tf = ((PiVi) / (RTi)) * (R / (PiVf))
Now we can calculate the final temperature:
Tf = (PiViTi) / (PiVf)
Let's plug in the given values to calculate the final temperature.
To find the final temperature of the ideal gas, you can use the combined gas law equation. The combined gas law is given by:
(P1 * V1) / T1 = (P2 * V2) / T2,
where P1, V1, and T1 are the initial pressure, volume, and temperature of the gas, and P2, V2, and T2 are the final pressure, volume, and temperature of the gas.
Let's substitute the given values into the equation:
(5 atm * 490 cm³) / (100°C) = (1 atm * 1000 cm³) / T2.
Now, let's solve for T2, the final temperature. Rearranging the equation gives:
T2 = (1 atm * 1000 cm³ * 100°C) / (5 atm * 490 cm³).
T2 = (100000 cm³°C) / (2450 cm³).
T2 = 40.816°C.
Therefore, the final temperature of the ideal gas is approximately 40.8°C.