Question

Calculate the osmotic pressure of a \(0.0120 \mathrm{M}\) solution of NaCl in water at \(0^{\circ}\) C. Assume the van't Hoff factor, \(i_{t}\) is 1.94 for this solution.

Short answer

The osmotic pressure is approximately 0.521 atm.

Step-by-step solution

Recall the Formula for Osmotic Pressure

The formula to calculate osmotic pressure \( \Pi \) is given by \[ \Pi = i_{t} \, M \, R \, T \] where \( i_{t} \) is the van 't Hoff factor, \( M \) is the molarity of the solution, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Convert Temperature to Kelvin

Given that the temperature is \(0^{\circ}\) C, convert this to Kelvin by adding 273.15. Therefore, \( T = 273.15 \) K.

Use the Given Values

Substitute the provided values into the osmotic pressure formula:- \( i_{t} = 1.94 \)- \( M = 0.0120 \) M- \( R = 0.0821 \) L·atm/(mol·K)- \( T = 273.15 \) KThis gives:\[ \Pi = 1.94 \times 0.0120 \, \text{M} \times 0.0821 \, \text{L·atm/(mol·K)} \times 273.15 \, \text{K} \]

Perform the Calculation

Compute the osmotic pressure by multiplying the values together:\[ \Pi = 1.94 \times 0.0120 \times 0.0821 \times 273.15 \approx 0.521 \text{ atm} \]

Key concepts

van't Hoff factor

The van't Hoff factor, often denoted as \( i_t \), is a crucial concept when calculating osmotic pressure in solutions. It is used to describe how many particles a solute forms in solution, which can affect various colligative properties. For instance, when a substance like sodium chloride (NaCl) dissolves, it typically dissociates into two ions, Na\(^+\) and Cl\(^-\).
This factor is therefore particularly important when dealing with ionic compounds. It reflects how much the presence of these ions influences the expected behavior of a solution, in this case, the osmotic pressure.

• If \( i_t \) is 1, it means no dissociation happens in the solution.
• A value greater than 1 indicates dissociation into multiple particles.

In our solution for NaCl in water, \( i_t = 1.94 \), suggesting slight discrepancies from the ideal dissociation into two ions due to interactions in the solution.

molarity

Molarity, symbolized as \( M \), quantifies the concentration of a solute in a solution. It is one of the most commonly used terms for expressing solution concentration in chemistry and is defined as the number of moles of solute per liter of solution.
This is why it's often written as mol/L or simply M.

• Molarity helps in understanding how concentrated a solution is, which directly impacts properties like osmotic pressure.
• For our NaCl solution, the molarity is given as \( 0.0120 \) M, meaning there are 0.0120 moles of NaCl per liter of water.

Knowing the molarity is essential for using the osmotic pressure formula, as it helps in calculating how much solute affects the solution's overall properties.

ideal gas constant

The ideal gas constant, denoted as \( R \), is a fundamental constant that appears in various theoretical constructs involving gases, such as the ideal gas law.
In the context of osmotic pressure, \( R \) is vital as it helps relate the molarity and temperature of the solution to its pressure.

• The value used in our calculation is \( R = 0.0821 \) L·atm/(mol·K), which is a common choice when pressure is expressed in atmospheres and volume in liters.
• This constant serves as a conversion factor that balances units in the osmotic pressure equation, ensuring molarity, temperature, and pressure interact correctly.

Understanding \( R \) and its role is essential for correctly applying the formula, which relates to how energy and temperature impact pressure in an ideal scenario.

temperature conversion

Temperature conversion is a critical step when solving for osmotic pressure in solutions, as the temperature must be expressed in Kelvin, the absolute temperature scale, to correctly use formulas in physical chemistry.
Celcius and Kelvin scales are related, and conversion is simple:

• The equation is \( T(K) = T(^{\circ}C) + 273.15 \).
• In our example, starting from \(0^{\circ}C\), this converts to \(273.15\) K.

Using Kelvin ensures that calculations related to gas laws and osmotic pressure align with the principles of thermodynamics, providing accurate results. Understanding and correctly applying temperature conversion ensures your calculations reflect how energy and molecular movement are impacting the system at hand.