# The compression and expansion of gases form the basis of how air is cooled by air conditioners. Suppose 1.55 L of an ideal gas under 6.38 atm of pressure at 20.5°C is expanded to 6.95 L at 1.00 atm. What is the new temperature?

Dec 25, 2015

$\text{206 K}$

#### Explanation:

The pressure, temperature, and volume of the gas will change when going from its initial state to its final state, which tells you that you must use the combined gas law to find the new temperature of the gas.

As you know, when the number of moles of gas is kept constant, pressure, volume, and temperature have the following relationship

$\textcolor{b l u e}{\frac{{P}_{1} {V}_{1}}{T} _ 1 = \frac{{P}_{2} {V}_{2}}{T} _ 2} \text{ }$, where

${P}_{1}$, ${V}_{1}$, ${T}_{1}$ - the pressure, volume, and temperature of the gas at an initial state
${P}_{2}$, ${V}_{2}$, ${T}_{2}$ - the pressure, volume, and temperature of the gas at a final state

From this point on, the important thing to remember is that the temperature of the gas must be expressed in Kelvin, so don't forget to convert it from the given degrees Celsius.

So, rearrange that equation to solve for ${T}_{2}$

${T}_{2} = {P}_{2} / {P}_{1} \cdot {V}_{2} / {V}_{1} \cdot {T}_{1}$

Plug in your values to get

T_2 = (1.00 color(red)(cancel(color(black)("atm"))))/(6.38color(red)(cancel(color(black)("atm")))) * (6.95 color(red)(cancel(color(black)("L"))))/(1.55color(red)(cancel(color(black)("L")))) * (273.15 + 20.5)"K"

${T}_{2} = \text{206.38 K}$

Rounded to three sig figs, the answer will be

${T}_{2} = \textcolor{g r e e n}{\text{206 K}}$

If you want, you can express the answer in degrees Celsius

T_2[""^@"C"] = 206 - 273.15 = -67.2^@"C"