Question

the universal gas law, pv = nrt, describes the relationship among the pressure, volume, and temperature of a gas. p = pressure v = volume t = temperature (kelvin) n = number of moles (quantity of gas particles) r = universal gas constant at a constant temperature of 1 k, the pressure and volume of 1 mole of gas vary inversely. use the values in the table to find the value of r.

volume (liters)	33.24	16.62	4.155
pressure (kilopascals)	0.25	0.5	2

Explanation:

Step1: Recall the ideal - gas law

The ideal - gas law is $PV = nRT$. We want to find $n$ (number of moles). Rearranging the formula for $n$, we get $n=\frac{PV}{RT}$.

Step2: Substitute the values from the table

Let's assume we take one set of values from the table. For example, if $P = 0.25$ kPa, $V = 33.24$ L, $R$ (universal gas constant), and $T$ (not given in the problem - but if we assume we are using the values in the context of finding $n$ for a given $P$ and $V$ with a known $R$). Let's assume $R$ is a constant value. If we consider the general form of using the values from the table to find $n$ for each pair of $P$ and $V$ values.
For the first row: $P = 0.25$ kPa, $V = 33.24$ L.
We need to know the value of $R$ and assume a temperature (if not considering temperature - dependent cases for a quick ratio calculation). If we assume standard conditions and use the appropriate value of $R$ (e.g., $R=8.314$ J/(mol·K) or in appropriate units for kPa and L). First, convert kPa to Pa ($1$ kPa = $1000$ Pa) and use the correct units for $R$. But if we just consider the values in the table in a non - SI unit context for a simple calculation of the ratio $\frac{PV}{R}$ (assuming $T = 1$ K for simplicity as we are not given $T$ and just showing the process of using the values from the table), and assume $R$ is a constant.
Let's assume $R$ is some non - zero constant. Then for the first pair:
$n_1=\frac{0.25\times33.24}{R\times1}=\frac{8.31}{R}$
For the second row: $P = 0.5$ kPa, $V = 16.62$ L, $n_2=\frac{0.5\times16.62}{R\times1}=\frac{8.31}{R}$
For the third row: $P = 2$ kPa, $V = 4.155$ L, $n_3=\frac{2\times4.155}{R\times1}=\frac{8.31}{R}$

We can see that the number of moles $n$ is the same for all the data points in the table when we use the ideal - gas law formula $n=\frac{PV}{RT}$ (assuming $T$ is constant).

If we assume $R$ is a standard value (e.g., $R = 8.314$ L·kPa/(mol·K)) and assume $T = 1$ K (for simplicity of using the values in the table), then $n=\frac{PV}{R}$.
For $P = 0.25$ kPa and $V = 33.24$ L:
$n=\frac{0.25\times33.24}{8.314\times1}=1$ mol
For $P = 0.5$ kPa and $V = 16.62$ L:
$n=\frac{0.5\times16.62}{8.314\times1}=1$ mol
For $P = 2$ kPa and $V = 4.155$ L:
$n=\frac{2\times4.155}{8.314\times1}=1$ mol

Answer:

The number of moles $n$ for each set of values in the table is $1$ mol (assuming $T = 1$ K and $R = 8.314$ L·kPa/(mol·K)).