To find the equation that agrees with the ideal gas law, let's start by understanding the Ideal Gas Law itself. The Ideal Gas Law is mathematically expressed as:
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] is the pressure of the gas,
- [tex]\( V \)[/tex] is the volume of the gas,
- [tex]\( n \)[/tex] is the number of moles of gas,
- [tex]\( R \)[/tex] is the universal gas constant, and
- [tex]\( T \)[/tex] is the temperature of the gas in Kelvin.
Now, suppose we have two different states of the same amount of gas. If the gas changes from state 1 to state 2, the values of pressure, volume, and temperature may change. For both states to agree with the ideal gas law, we can write the expressions for those states as:
[tex]\[ P_1 V_1 = n R T_1 \][/tex] (for State 1)
[tex]\[ P_2 V_2 = n R T_2 \][/tex] (for State 2)
Since the amount of gas [tex]\( n \)[/tex] and the gas constant [tex]\( R \)[/tex] are the same in both states, we can set up a comparison between the two states. By rearranging the equations, we equate them to form:
[tex]\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \][/tex]
To make it clear, this equation states that the ratio of the product of pressure and volume to the temperature for one state must be equal to the same ratio for another state. This is an expression of consistency with the ideal gas law when comparing two states of the same gas.
If we further rearrange this relation to bring all terms to one side of the equation, it takes the form:
[tex]\[ \frac{P_1 V_1}{T_1} - \frac{P_2 V_2}{T_2} = 0 \][/tex]
This represents that the differences between the ratios of pressure and volume to temperature in two states will be zero when the gas obeys the ideal gas law.
Thus, the equation:
[tex]\[ \frac{P_1 V_1}{T_1} - \frac{P_2 V_2}{T_2} \][/tex]
agrees with the ideal gas law when comparing the states of the gas.
