Gaseous State

To determine the correct unit for the van der Waals constant $a$ , we need to analyze its role and the units involved in the van der Waals equation of state for real gases. The van der Waals equation is an improvement over the ideal gas law, accounting for intermolecular forces and the finite size of gas molecules. $%5Cleft(%20P%20%2B%20%5Cfrac%7Ba%7D%7BV_m%5E2%7D%20%5Cright)(V_m%20-%20b)%20%3D%20RT$ Where:

$P$ : Pressure
$V_m$ : Molar volume
$T$ : Temperature
$R$ : Universal gas constant
$a$ : van der Waals constant accounting for intermolecular forces
$b$ : van der Waals constant accounting for the finite size of molecules

Step 1: Understand the Units of Each Term
First, let’s identify the units of each variable in the equation:
- Pressure ( $P$ ) has units of atmospheres (atm).
- Molar volume ( $V_m$ ) has units of cubic decimeters per mole ( $%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D$ ).
- Temperature ( $T$ ) has units of Kelvin (K).
- Gas constant ( $R$ ) has units of $%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Batm%7D%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D%20%5C%2C%20%5Ctext%7BK%7D%5E%7B-1%7D$ .
Step 2: Analyze the Term Involving $a$
The term $%5Cfrac%7Ba%7D%7BV_m%5E2%7D$ is added to the pressure $P$ . For dimensional consistency, the units of $%5Cfrac%7Ba%7D%7BV_m%5E2%7D$ must be the same as those of $P$ , which is atm.
$%5Cfrac%7Ba%7D%7BV_m%5E2%7D%20%5C%2C%20%5Ctext%7Bhas%20units%20of%7D%20%5C%2C%20%5Ctext%7Batm.%7D$
Step 3: Express $a$ in Terms of Units
Given that: $%5Cfrac%7Ba%7D%7BV_m%5E2%7D%20%5C%2C%20%5Ctext%7Bhas%20units%20of%7D%20%5C%2C%20%5Ctext%7Batm%2C%7D$ we can express the units of $a$ as: $%5Ctext%7BUnits%20of%7D%20%5C%2C%20a%20%3D%20%5Ctext%7Batm%7D%20%5Ctimes%20(%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D)%5E2%20%3D%20%5Ctext%7Batm%7D%20%5Ctimes%20%5Ctext%7Bdm%7D%5E6%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-2%7D.$ $%5Ctext%7BUnits%20of%7D%20%5C%2C%20a%20%3D%20%5Ctext%7Bdm%7D%5E6%20%5C%2C%20%5Ctext%7Batm%7D%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-2%7D.$
Step 4: Confirm Dimensional Consistency in the van der Waals Equation
To ensure our derived units for are correct, let’s check the dimensional consistency of the entire equation:
- $P$ has units of atm.
- $%5Cfrac%7Ba%7D%7BV_m%5E2%7D$ has units of atm.
- $V_m$ and $b$ both have units of $%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D$ .
- $RT$ has units of $%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Batm%7D%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D$ (since $R$ is in $%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Batm%7D%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D%20%5C%2C%20%5Ctext%7BK%7D%5E%7B-1%7D$ and $T$ is in K).

Multiplying the terms on the left side: $%5Ctext%7Batm%7D%20%5Ctimes%20%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D%20%3D%20%5Ctext%7Bdm%7D%5E3%20%5C%2C%20%5Ctext%7Batm%7D%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-1%7D.$ Which matches the units on the right side ( $RT$ ), confirming dimensional consistency.
Conclusion:
The van der Waals constant $a$ must have units of $%5Ctext%7Bdm%7D%5E6%20%5C%2C%20%5Ctext%7Batm%7D%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-2%7D$ to ensure that the van der Waals equation remains dimensionally consistent. Therefore, the correct unit for $a$ is: $%5Cboxed%7B%5Ctext%7Bdm%7D%5E6%20%5C%2C%20%5Ctext%7Batm%7D%20%5C%2C%20%5Ctext%7Bmol%7D%5E%7B-2%7D%7D$

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Gaseous State

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs