Question

all pressure–volume–temperature relationships for gases can be combined into a single relationship known as the combined gas law. this expression can be used when looking at the effect of changes in two of these variables on the third as long as the amount of gas (number of moles) remains constant. to use the combined gas law properly, you must always express the temperatures in kelvins. the combined gas law can be represented as follows.
\\(\frac{p_1v_1}{t_1} = \frac{p_2v_2}{t_2}\\)

air pressure decreases as you move higher into the atmosphere. the decrease in pressure encourages the volume of the balloon to increase by quite a bit, whereas the drop in temperature will encourage some decrease in the volume. so in this case, the pressure and temperature play opposite roles in affecting the volume.

part b
consider 5.00 l of a gas at 365 mmhg and 20. °c. if the container is compressed to 2.80 l and the temperature is increased to 40. °c, what is the new pressure, \\(p_2\\), inside the container? assume no change in the amount of gas inside the cylinder.
express your answer with the appropriate units.
view available hint(s)

Explanation:

Step1: Convert temperatures to Kelvin

Convert \( T_1 = 20^\circ\text{C} \) to Kelvin: \( T_1 = 20 + 273.15 = 293.15\,\text{K} \)
Convert \( T_2 = 40^\circ\text{C} \) to Kelvin: \( T_2 = 40 + 273.15 = 313.15\,\text{K} \)

Step2: Identify known values

\( P_1 = 365\,\text{mmHg} \), \( V_1 = 5.00\,\text{L} \), \( V_2 = 2.80\,\text{L} \), \( T_1 = 293.15\,\text{K} \), \( T_2 = 313.15\,\text{K} \)

Step3: Rearrange combined gas law for \( P_2 \)

From \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \), solve for \( P_2 \):
\( P_2 = P_1 \cdot \frac{V_1}{V_2} \cdot \frac{T_2}{T_1} \)

Step4: Substitute values and calculate

Substitute values:
\( P_2 = 365\,\text{mmHg} \cdot \frac{5.00\,\text{L}}{2.80\,\text{L}} \cdot \frac{313.15\,\text{K}}{293.15\,\text{K}} \)
Calculate \( \frac{5.00}{2.80} \approx 1.7857 \), \( \frac{313.15}{293.15} \approx 1.0682 \)
Multiply: \( 365 \cdot 1.7857 \cdot 1.0682 \approx 365 \cdot 1.907 \approx 696\,\text{mmHg} \)

Answer:

\( P_2 \approx \boxed{696\,\text{mmHg}} \)