At 10°C, the gas in a cylinder has a volume of 0.250 L. The gas is allowed to expand to 0.285 L. What must the final temperature be for the pressure to remain constant?

Question

Avery Adams · Accepted Answer

#50˚"C"# Use the formula: #(P_1V_1)/(T_1)=(P_2V_2)/(T_2)# We know that we want #P_1=P_2# so the pressure will remain constant, so we can say that: #(color(blue)(P_1)V_1)/(T_1)=(color(blue)(P_1)V_2)/(T_2)# Plug in the values we know: #(P_1*0.250"L")/(283"K")=(P_1*0.285"L")/(T_2)# (Remember that temperature must be done in Kelvin.) Cross multiply: #T_2*P_1*0.250"L"=283"K"*P_1*0.285"L"# Divide both sides by #P_1#. #T_2*0.250"L"=283"K"*0.285"L"# #T_2=(283"K"*0.285"L")/(0.250"L")# #T_2=323"K"# (This is also #50˚"C"#.)

Noah Aldrich · Answer

undefined To find the final temperature, you can use the formula for Charles's Law:

$ \frac{V_1}{T_1} = \frac{V_2}{T_2} $

Given:
$ V_1 = 0.250 \, \text{L} $
$ V_2 = 0.285 \, \text{L} $
$ T_1 = 10^\circ \text{C} + 273.15 = 283.15 \, \text{K} $

Substituting the values into the formula:

$ \frac{0.250}{283.15} = \frac{0.285}{T_2} $

Solving for $ T_2 $:

$ T_2 = \frac{0.285 \times 283.15}{0.250} $

$ T_2 \approx 322.19 \, \text{K} $

Converting back to Celsius:

$ T_2 = 322.19 - 273.15 $

$ T_2 \approx 49.04^\circ \text{C} $

So, the final temperature should be approximately $ 49.04^\circ \text{C} $ for the pressure to remain constant.