# Question a7aeb

Jun 8, 2016

$\text{600 torr}$

#### Explanation:

The first thing to recognize here is that no mention is made of volume and number of moles of gas, which means that you can safely assume that they are being kept constant.

Under these conditions, pressure and temperature have a direct relationship described by Gay Lussac's Law.

In simply terms, when temperature increases, pressure increases as well, and when pressure decreases, temperature decreases as well.

Mathematically, this is expressed as

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {P}_{1} / {T}_{1} = {P}_{2} / {T}_{2} \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$, where

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

In your case, the temperature is said to go from $\text{300 K}$ to $\text{450 K}$. This increase in temperature could have only resulted from an increase in pressure, so right from the start you know that

${P}_{1} < \text{900 torr} \to$.the pressure of the gas increased

Rearrange the above equation to solve for ${P}_{1}$

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

Plug in your values to find

${P}_{1} = \left(300 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{K"))))/(450color(red)(cancel(color(black)("K")))) * "900 torr" = color(green)(|bar(ul(color(white)(a/a)color(black)("600 torr}} \textcolor{w h i t e}{\frac{a}{a}} |}}\right)$

As predicted, the pressure of the gas increased from $\text{600 K}$ to $\text{900 K}$, which in turn caused the temperature of the gas to increase from $\text{300 K}$ to $\text{450 K}$.

SIDE NOTE The equation that describes Gay Lussac's Law can be derived from the ideal gas law equation

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} P V = n R T \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$

where

$P$ - the pressure of the gas
$V$ - the volume it occupies
$n$ - the number of moles of gas
$R$ - the universal gas constant
$T$ - the absolute temperature of the gas

When volume and number of moles are kept constant, you can rearrange the above equation to isolate the constants on one side of the equation

$P V = n R T \implies \frac{P}{T} = {\overbrace{\frac{n R}{V}}}^{\textcolor{red}{\text{constant}}}$

This tells you that under these conditions, pressure and temperature have a direct relationship, i.e. when one increases, the other must increase by the same factor in order to keep the $\frac{P}{T}$ ratio constant.

Therefore, if you have a gas at ${P}_{1}$ and ${T}_{1}$, and then at a second state ${P}_{2}$ and ${T}_{2}$, you will have

P_1/T_1 = color(red)("constant") " " and " " P_2/T_2 = color(red)("constant")#

which implies that

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {P}_{1} / {T}_{1} = {P}_{2} / {T}_{2} \textcolor{w h i t e}{\frac{a}{a}} |}}} \to$ the equation that describes Gay Lussac's Law