To determine which equation agrees with the ideal gas law, let’s analyze each option step by step in accordance with the ideal gas law, which is expressed as:
[tex]\[ PV = nRT \][/tex]
where [tex]\( P \)[/tex] stands for pressure, [tex]\( V \)[/tex] for volume, [tex]\( n \)[/tex] for the number of moles of the gas, [tex]\( R \)[/tex] for the universal gas constant, and [tex]\( T \)[/tex] for temperature.
Given three choices:
1. [tex]\( V₁ = V₂ \)[/tex]
2. [tex]\( P₁ = P₂ \)[/tex]
3. [tex]\( \frac{P₁}{T₁} = \frac{P₂}{T₂} \)[/tex]
We will assess whether each equation aligns with the ideal gas law principles.
### Option 1: [tex]\( V₁ = V₂ \)[/tex]
This equation states that the volumes [tex]\( V₁ \)[/tex] and [tex]\( V₂ \)[/tex] are equal. However, this doesn’t directly relate to the pressure or temperature variables. Given the ideal gas law [tex]\( PV = nRT \)[/tex], volume could only be constant if the product of the pressure and the temperature (assuming the number of moles and gas constant remain unchanged) remains equivalent, but this option alone doesn’t encompass the relationship needed.
### Option 2: [tex]\( P₁ = P₂ \)[/tex]
This equation implies that the pressures [tex]\( P₁ \)[/tex] and [tex]\( P₂ \)[/tex] are equal. Similar to option 1, this does not account for the potential variation in volumes or temperatures. According to [tex]\( PV = nRT \)[/tex], pressure can stay constant only if a balance is maintained with changes in volume and temperature, but this agreement alone is insufficient for ideal gas behavior explanation.
### Option 3: [tex]\( \frac{P₁}{T₁} = \frac{P₂}{T₂} \)[/tex]
This equation can be rearranged for clarity to [tex]\( P₁ T₂ = P₂ T₁ \)[/tex]. This equation indicates that the pressures and temperatures of two states of the gas are related proportionally, which makes more sense in the context of the ideal gas law because it inherently considers alternate scenarios with pressures and temperatures, holding volume and the number of moles constant.
Therefore, the option that reflects the correct proportional relationship according to the ideal gas law is:
[tex]\[ \frac{P₁}{T₁} = \frac{P₂}{T₂} \][/tex]
Thus, the correct equation that agrees with the ideal gas law is:
[tex]\[ \boxed{3} \][/tex]
