A gas occupies #"67. cm"^3# at #9.38 × 10^4color(white)(l)"Pa"# and 22 °C. What is its volume at #10.6 × 10^5color(white)(l)"Pa"# and 29 °C?
So, it's always a good idea to start by making a note of what information is being provided by the problem.
In your case, you know that the initial sample of gas
- occupies a volume equal to
- has a temperature of
- has a pressure of
#9.38 * 10^4"Pa"#
You then go on to change the temperature to
Notice that no mention of number of moles was made. This means that you can assume it to be constant. So, if you start from the ideal gas law equation, you can say that
#P_1 * V_1 = n * R * T_1 ->#the initial state of the gas
#P_2 * V_2 = n * R * T_2 ->#the final state of the gas
#(P_1 * V_1)/T_1 = n * R" "#and #" "(P_2 * V_2)/T_2 = n * R#
Notice that you have two expressions that are equal to the same value,
#(P_1 * V_1)/T_1 = (P_2 * V_2)/T_2 ->#the combined gas law equation
Now all you have to do is rearrange this to solve for
Look what happens if you divide both sides of the equation by
#P_1/P_2 * V_1/T_1 = (color(red)(cancel(color(black)(P_2))) * V_2)/(T_2 * color(red)(cancel(color(black)(P_2))))#
#P_1/P_2 * V_1/T_1 = V_2/T_2#
Now multiply both sides by
#P_1/P_2 * T_2/T_1 * V_1 = V_2/color(red)(cancel(color(black)(T_2))) * color(red)(cancel(color(black)(T_2)))#
Finally, you got
#V_2 = P_1/P_2 * T_2/T_1 * V_1#
Now plug in your values and solve for
#V_2 = (9.38 * 10^4color(red)(cancel(color(black)("Pa"))))/(10.6 * 10^5color(red)(cancel(color(black)("Pa")))) * ((273.15 + 29)color(red)(cancel(color(black)("K"))))/((273.15 + 22)color(red)(cancel(color(black)("K")))) * "67 cm"^3#
#V_2 = "6.0695 cm"^3#
You need to round this off to two sig figs, the number of sig figs you have for the initial volume of the gas
#V_2 = color(green)("6.1 cm"^3)#
The new volume will be
We use the Combined Gas Law equation,
Let's start by listing our given information.
Now we must rearrange the Combined Gas Law equation to get
We'll take it step by step.
Step 1. Multiply both sides by
Step 2. Divide both sides by
Now we insert the values into the equation.
Check: The temperature doesn't change much, but the pressure increases by about ten-fold.
The new volume should be about one-tenth of the original volume, or about