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)	4.155	16.62	33.24

Explanation:

Step1: Recall the universal gas law formula

The universal gas law is given by \( pV = nRT \). We need to solve for \( R \), so we can rearrange the formula to \( R=\frac{pV}{nT} \).

Step2: Identify the values for each variable

From the problem, we know that \( T = 1\space K \) (constant temperature), \( n = 1\space mole \) (since it's 1 mole of gas), and we can take one set of \( p \) and \( V \) values. Let's take the first set: \( p = 2\space kilopascals \), \( V = 4.155\space liters \), \( n = 1 \), \( T = 1 \).

Step3: Substitute the values into the formula for \( R \)

Substituting \( p = 2 \), \( V = 4.155 \), \( n = 1 \), and \( T = 1 \) into \( R=\frac{pV}{nT} \), we get:
\( R=\frac{2\times4.155}{1\times1} \)
\( R = 8.31 \)? Wait, no, wait the table: Wait, maybe I misread. Wait the table has three rows? Wait the first row: Volume (Liters): 4.155, Pressure (kilopascals): 2; second row: Volume 16.62, Pressure 0.5; third row: Volume 33.24, Pressure 0.25. And \( n = 1 \) mole, \( T = 1 \) K. Let's check with the first row: \( p = 2 \), \( V = 4.155 \), \( n = 1 \), \( T = 1 \). Then \( R=\frac{pV}{nT}=\frac{2\times4.155}{1\times1}=8.31 \)? Wait but let's check the second row: \( p = 0.5 \), \( V = 16.62 \), \( n = 1 \), \( T = 1 \). Then \( R=\frac{0.5\times16.62}{1\times1}=8.31 \). Third row: \( p = 0.25 \), \( V = 33.24 \), \( n = 1 \), \( T = 1 \). Then \( R=\frac{0.25\times33.24}{1\times1}=8.31 \). Ah, so regardless of the row, since \( p \) and \( V \) are inversely proportional (as given \( p=\frac{R}{V} \) when \( n = 1 \), \( T = 1 \)), so \( R = pV \). Let's take the first row: \( p = 2 \), \( V = 4.155 \), so \( R = 2\times4.155 = 8.31 \). Second row: \( 0.5\times16.62 = 8.31 \). Third row: \( 0.25\times33.24 = 8.31 \). So \( R = 8.31 \). Wait, but let's confirm the formula. The universal gas law is \( pV = nRT \), so solving for \( R \), \( R=\frac{pV}{nT} \). Given \( n = 1 \) (1 mole), \( T = 1 \) K, so \( R = pV \). So using any of the rows, we get \( R = 8.31 \). Let's check the calculations:

First row: \( p = 2 \), \( V = 4.155 \), so \( 2\times4.155 = 8.31 \).

Second row: \( 0.5\times16.62 = 8.31 \) (since \( 16.62\times0.5 = 8.31 \)).

Third row: \( 0.25\times33.24 = 8.31 \) (since \( 33.24\times0.25 = 8.31 \)).

So \( R = 8.31 \).

Answer:

\( 8.31 \)