Answer :
To determine the equation that agrees with the ideal gas law, we can start by analyzing the principles behind the Ideal Gas Law and derive the pertinent relationships.
The ideal gas law is typically expressed as:
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] represents the pressure of the gas
- [tex]\( V \)[/tex] represents the volume of the gas
- [tex]\( n \)[/tex] represents the number of moles of the gas
- [tex]\( R \)[/tex] is the universal gas constant
- [tex]\( T \)[/tex] represents the temperature (in Kelvin)
For conditions of constant moles and constant gas constant [tex]\( R \)[/tex], we can rearrange the ideal gas equation to compare different states of a gas. Hence, if we have initial state 1 and final state 2, the equation can be written as:
[tex]\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \][/tex]
Now, let's solve for one of the volumes, here using [tex]\( V_1 \)[/tex]:
[tex]\[ V_1 = \frac{P_2 V_2 T_1}{P_1 T_2} \][/tex]
Now, considering that we need an equation that can answer the given format, we use algebraic manipulation:
[tex]\[ \frac{V_1}{T_1} = \frac{P_2 V_2}{P_1 T_2} \][/tex]
Rewriting, we obtain:
[tex]\[ \frac{V_1}{T_1} \cdot P_2 = \frac{P_2 V_2}{T_1} = P_1 T_2 \][/tex]
So, after simplifying further:
[tex]\[ (V_1 / T_1) \cdot P_2 = P_1 \cdot T_2 \][/tex]
Thus, the equation that conforms to the ideal gas law is:
[tex]\[ (V_1 / T_1) \cdot P_2 = P_1 \cdot T_2 \][/tex]
The ideal gas law is typically expressed as:
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] represents the pressure of the gas
- [tex]\( V \)[/tex] represents the volume of the gas
- [tex]\( n \)[/tex] represents the number of moles of the gas
- [tex]\( R \)[/tex] is the universal gas constant
- [tex]\( T \)[/tex] represents the temperature (in Kelvin)
For conditions of constant moles and constant gas constant [tex]\( R \)[/tex], we can rearrange the ideal gas equation to compare different states of a gas. Hence, if we have initial state 1 and final state 2, the equation can be written as:
[tex]\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \][/tex]
Now, let's solve for one of the volumes, here using [tex]\( V_1 \)[/tex]:
[tex]\[ V_1 = \frac{P_2 V_2 T_1}{P_1 T_2} \][/tex]
Now, considering that we need an equation that can answer the given format, we use algebraic manipulation:
[tex]\[ \frac{V_1}{T_1} = \frac{P_2 V_2}{P_1 T_2} \][/tex]
Rewriting, we obtain:
[tex]\[ \frac{V_1}{T_1} \cdot P_2 = \frac{P_2 V_2}{T_1} = P_1 T_2 \][/tex]
So, after simplifying further:
[tex]\[ (V_1 / T_1) \cdot P_2 = P_1 \cdot T_2 \][/tex]
Thus, the equation that conforms to the ideal gas law is:
[tex]\[ (V_1 / T_1) \cdot P_2 = P_1 \cdot T_2 \][/tex]
