Question

Analyzing Information The osmotic pressure of a dilute solution can be calculated as follows: \(\pi=M R T\) \(\pi=\) osmotic pressure \(M=\) concentration in moles per liter \(R=\) ideal gas constant \(T=\) absolute temperature of the solution How does the osmotic-pressure equation compare with the ideal gas law?

Short answer

The osmotic pressure equation \( \pi = M R T \)is analogous to the ideal gas law because both relate pressure to moles or concentration, the gas constant, and temperature.

Step-by-step solution

- Identify the given equation

The given equation for osmotic pressure is \( \pi = M R T \) where \( \pi \) is osmotic pressure, \( M \) is concentration in moles per liter, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature.

- Recall the ideal gas law

The ideal gas law is given by \( PV = nRT \) where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature.

- Compare the two equations

Compare the osmotic pressure equation \( \pi = M R T \) with the ideal gas law \( PV = nRT \). To compare these, note that osmotic pressure \( \pi \) corresponds to the gas pressure \( P \), and molarity \( M \) (moles per liter) corresponds to \( \frac{n}{V} \) (which is moles per volume L).

- Identify similarities

Both equations involve a pressure term (\( \pi \) and \( P \)), the ideal gas constant \( R \), and the temperature \( T \). Osmotic pressure \( \pi \) behaves similarly to the pressure \( P \) in the ideal gas law when the concentration \( M \) of a dilute solution is analogous to \( \frac{n}{V} \).

- Summarize the comparison

The osmotic-pressure equation \( \pi = M R T \) is analogous to the ideal gas law \( PV = nRT \) because both relate a type of pressure to the product of the number of particles (whether in moles or concentration), the ideal gas constant, and temperature.

Key concepts

Ideal Gas Law

The **ideal gas law** is a fundamental equation in chemistry and physics. It connects the pressure, volume, and temperature of an ideal gas. The equation is represented as:

\( PV = nRT \).

Here:

• \(P\) is the pressure of the gas
• \(V\) is the volume occupied by the gas
• \(n\) is the number of moles of the gas
• \(R\) is the ideal gas constant
• \( T \) is the absolute temperature

This law helps us understand how changing one property (like temperature) affects the others (like pressure or volume). It is crucial in predicting how gases will behave under different conditions.

Concentration

In the context of the osmotic pressure equation, **concentration** refers to the amount of solute (in moles) present in one liter of solution. It is represented by \( M \) and is measured in moles per liter (M or mol/L).

Concentration is critical because it directly affects the osmotic pressure. More solute particles in a solution create a higher osmotic pressure. This concept is similar to the number of moles per volume (\( \frac{n}{V} \)) in the ideal gas law.

High concentration means more particles are available to exert pressure. In the osmotic pressure equation \( \pi = MRT \), \( M \) tells us how concentrated the solution is, influencing the total pressure \( \pi \).

Temperature

The **temperature** of a solution is another critical factor. It is denoted by \( T \) and always measured in Kelvin for these equations.

In both the ideal gas law and the osmotic pressure equation, temperature plays a vital role. According to the equations:

• If the temperature increases, the pressure also increases, assuming other variables stay constant.
• A higher temperature means higher energy for gas particles or solute particles, leading to more collisions and thus higher pressure.

Always remember to convert Celsius to Kelvin by adding 273.15 to the Celsius temperature before using it in these equations.

Ideal Gas Constant

The **ideal gas constant**, represented by \( R \), is a universal constant. Its value is approximately \( 0.0821 \) L⋅atm/(K⋅mol) or \( 8.314 \) J/(K⋅mol).

This constant appears in both the ideal gas law and the osmotic pressure equation. It bridges the relationship between pressure, volume, molarity, and temperature.

Understand that using the right units is essential. For example, if using \( 0.0821 \), pressure must be in atmospheres, volume in liters, and temperature in Kelvin. This constant ensures consistency and accuracy in calculations.