Question

A 0.86 percent by mass solution of $\mathrm{NaCl}$ is called "physiological saline" because its osmotic pressure is equal to that of the solution in blood cells. Calculate the osmotic pressure of this solution at normal body temperature $\left({37}^{\circ }\mathrm{C}\right).$ Note that the density of the saline solution is $1.005\mathrm{g}/\mathrm{mL}$.

Short answer

The osmotic pressure is approximately 7.53 atm.

Step-by-step solution

Identify the Formula

To find the osmotic pressure ($\pi$), we use the formula $\pi =iMRT$, where $i$ is the van 't Hoff factor, $M$ is the molarity of the solution, $R$ is the ideal gas constant ($0.0821{\text{L atm mol}}^{-1}{\text{K}}^{-1}$), and $T$ is the temperature in Kelvin.

Convert Temperature to Kelvin

Convert 37°C to Kelvin by adding 273.15 to it: $T=37+273.15=310.15\text{K}$.

Calculate Molarity of NaCl

First, find the mass of NaCl in 1 L of solution. Since the density is $1.005\text{g/mL}$, 1 L of solution weighs 1005 g. With a 0.86% mass concentration, the mass of NaCl is $1005\text{g}\times 0.0086=8.643\text{g}$.$$\text{Moles of NaCl}=\frac{8.643\text{g}}{58.44\text{g/mol}}=0.148\text{mol}\text{ (using NaCl molar mass}58.44\text{g/mol).}$$ So, molarity $M=0.148\text{mol/L}$.

Consider the Van 't Hoff factor

For NaCl, which dissociates completely into Na⁺ and Cl⁻ ions in solution, the van 't Hoff factor $i$ is 2, since $extNaCl\to ext{Na}^{+}+ext{Cl}^{-}$.

Calculate Osmotic Pressure

Substitute the values into the osmotic pressure formula: $\pi =iMRT$.$$\pi =2\times 0.148\text{mol/L}\times 0.0821{\text{L atm mol}}^{-1}{\text{K}}^{-1}\times 310.15\text{K}=7.528\text{atm}$$

Conclusion

The osmotic pressure of the physiological saline solution at 37°C is approximately 7.53 atm.

Key concepts

Physiological Saline

Physiological saline is a solution that closely matches the concentration of solutes found in body fluids, most notably in blood cells. It typically contains 0.86% sodium chloride (NaCl) by mass, which is equivalent to the salt concentration in our blood plasma. Physiological saline is often used in medical applications to hydrate the body and administer medications without causing harm to the cells by osmotic pressure changes.

The concept of matching the osmotic pressure is crucial because an incorrect balance can cause cells to either swell and burst or shrink and become damaged. The balanced osmotic pressure between the saline solution and blood cells prevents this harmful effect, making physiological saline a safe and effective option for medical treatments.

Van 't Hoff Factor

The van 't Hoff factor, denoted as $i$, is a measure of the effect of a solute on colligative properties, such as osmotic pressure. This factor accounts for the number of particles into which a solute dissociates in solution. For example, NaCl in water dissociates completely into two ions: Na⁺ and Cl⁻.

The van 't Hoff factor for NaCl is therefore $i=2$. Knowing this helps accurately calculate changes such as osmotic pressure, as it directly impacts the formula $\pi =iMRT$. This factor is crucial in predicting how solutes will behave in solution, particularly in important biological contexts or industrial applications where precise pressures are essential.

Molarity Calculation

Molarity, represented by $M$, is the measure of the concentration of a solute in a solution, expressed as moles of solute per liter of solution. Calculating molarity involves determining the amount of solute (in moles) and the total volume of the solution (in liters).

In the case of our physiological saline:

• We start with the mass percent of NaCl, which is 0.86%. This means 0.86 grams of NaCl per 100 grams of the solution.
• Using the solution's density (1.005 g/mL), we find that 1 liter (1000 mL) weighs 1005 grams.
• The mass of NaCl in 1 liter is then 8.643 grams (1005 grams * 0.0086).
• Next, we convert this mass into moles using the molar mass of NaCl, which is 58.44 g/mol, resulting in 0.148 moles of NaCl.

Thus, the molarity $M$ is 0.148 mol/L.

Temperature Conversion

When dealing with scientific calculations, it's essential to use the Kelvin scale for temperature, especially in gas-related equations like the one for osmotic pressure. The Kelvin scale is an absolute temperature scale that starts at absolute zero, making it ideal for use in physical science.

To convert a Celsius temperature to Kelvin, simply add 273.15. For physiological saline at body temperature:

• Start with 37°C.
• Add 273.15 to obtain Kelvin.
• This gives us 310.15 K.

This conversion is crucial as it ensures consistency in calculations that involve the ideal gas constant, $R$, which is defined in terms of Kelvin.

Solute Concentration

Solute concentration is a key component in understanding how solutions behave, particularly with respect to colligative properties like osmotic pressure. The concentration of a solute determines how many solute particles are present, which in turn affects how the solution will interact with other substances or barriers, such as cell membranes.

For the physiological saline solution, understanding its solute concentration allows us to accurately replicate the environment found in blood cells. With a NaCl concentration of 0.86% by mass, the calculation steps showed us how to convert this into molarity (0.148 mol/L), which is necessary for further calculations of osmotic pressure using the formula $\pi =iMRT$.

This concentration ensures the solution is isotonic to blood, preventing any osmosis-induced damage to cells.