To determine which statement is the most useful for deriving the ideal gas law, we need to recall the form and components of the ideal gas law:
The Ideal Gas Law is given by:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure,
- [tex]\( V \)[/tex] is the volume,
- [tex]\( n \)[/tex] is the number of moles of the gas,
- [tex]\( R \)[/tex] is the ideal gas constant,
- [tex]\( T \)[/tex] is the temperature in Kelvin.
The key task is to identify which of the given statements aligns correctly with this relationship.
Let's break down each statement:
1. Volume is directly proportional to the number of moles.
- According to the ideal gas law [tex]\[ V = \frac{nRT}{P} \][/tex].
- This illustrates that if [tex]\( P \)[/tex] (pressure) and [tex]\( T \)[/tex] (temperature) are constant, [tex]\( V \)[/tex] (volume) is directly proportional to [tex]\( n \)[/tex] (number of moles).
- This is correct because increasing the amount of gas (in moles) increases the volume, assuming pressure and temperature are constant.
2. Volume is inversely proportional to the temperature.
- According to [tex]\[ V = \frac{nRT}{P} \][/tex].
- This indicates that [tex]\( V \)[/tex] is directly proportional to [tex]\( T \)[/tex] when [tex]\( P \)[/tex] and [tex]\( n \)[/tex] are constant, which means this statement is incorrect.
- This statement misinterprets the direct relationship between volume and temperature.
3. Pressure is directly proportional to the volume.
- Using [tex]\( PV = nRT \)[/tex], rearranging as [tex]\( P = \frac{nRT}{V} \)[/tex], indicates that [tex]\( P \)[/tex] is inversely proportional to [tex]\( V \)[/tex] when [tex]\( n \)[/tex] and [tex]\( T \)[/tex] are constant.
- This statement is incorrect since it states a direct proportionality when, in reality, the relation is inverse.
4. Pressure is inversely proportional to the number of moles.
- Using [tex]\( P = \frac{nRT}{V} \)[/tex], we see [tex]\( P \)[/tex] is directly proportional to [tex]\( n \)[/tex] when [tex]\( V \)[/tex] and [tex]\( T \)[/tex] are constant.
- This statement is incorrect because it indicates an inverse proportionality instead of a direct relationship.
From the analysis above, the most useful statement for deriving the ideal gas law is indeed:
- Volume is directly proportional to the number of moles.
Thus, the correct choice is:
Volume is directly proportional to the number of moles.
