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.
$p = \frac{k}{v}$
use the values in the table to find the value of $r$.

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

$r = \square$

Explanation:

Step1: Recall the universal gas law formula

The universal gas law is given by \( pV = nRT \). We know that \( n = 1 \) mole, \( T = 1 \) K (constant temperature), and we have values of \( p \) (pressure) and \( V \) (volume) from the table. We can solve for \( R \) by rearranging the formula: \( R=\frac{pV}{nT} \).

Step2: Substitute values for the first row

For the first row, \( p = 2 \) kilopascals, \( V = 4.155 \) liters, \( n = 1 \), \( T = 1 \). Substitute into the formula: \( R=\frac{2\times4.155}{1\times1}=\frac{8.31}{1} = 8.31 \).

Step3: Verify with the second row

For the second row, \( p = 0.5 \) kilopascals, \( V = 16.62 \) liters. Substitute into \( R=\frac{pV}{nT} \): \( R=\frac{0.5\times16.62}{1\times1}=\frac{8.31}{1}=8.31 \).

Step4: Verify with the third row

For the third row, \( p = 0.25 \) kilopascals, \( V = 33.24 \) liters. Substitute into \( R=\frac{pV}{nT} \): \( R=\frac{0.25\times33.24}{1\times1}=\frac{8.31}{1}=8.31 \).

Answer:

\( 8.31 \) (with appropriate units, e.g., \( \text{kPa·L/(mol·K)} \))