Answer
a) $V(T)=0.0020525 T$
b) $0.513$ liters
c) $365.41K$
Work Step by Step
a) Let $V(T)$ be the volume in liters needed to store 0.05 mol of helium at a temperature $T$ in Kelvin. Since this volume varies directly with the temperature, we may write
\begin{equation}
\begin{aligned}
V(h)&= kT,
\end{aligned}
\end{equation} where $k$ is the constant of proportionality that must be determined. Given that $V(400)= 0.821$ liters when $T=400$ Kelvin, we can use this information to find $k$ as follows:
\begin{equation}
\begin{aligned}
0.821&= 400k\\
\frac{0.821}{400}&= k\\
0.0020525&= k.
\end{aligned}
\end{equation} We can write
\begin{equation}
\begin{aligned}
V(T)&= 0.0020525T.
\end{aligned}
\end{equation} b) Find $V(250)$ when $T= 250$.
\begin{equation}
\begin{aligned}
V(250)&=0.0020525\cdot 250\\
&= 0.513125.
\end{aligned}
\end{equation} The volume needed to store $0.05$ mol of helium at a temperature $250K$ is around $0.513$ liters.
c) Set $V(T) = 0.75$ and solve for $T$.
\begin{equation}
\begin{aligned}
V(T)&=0.75\\
0.0020525T &=0.75\\
T&= \frac{0.75}{0.0020525}\\
& \approx 365.41
\end{aligned}
\end{equation} The temperature required to store $0.05$ mol of helium at a volume of $0.75$ liter is about $365.41K$.
