Step 1: Understanding the Concept:
For a medical solution to be safely injected into the bloodstream, it must be isotonic with the blood plasma.
Isotonic solutions have the same osmotic pressure as the internal fluid of red blood cells.
If a hypertonic solution (higher pressure) is injected, water would flow out of the cells, causing them to shrink (crenation).
If a hypotonic solution (lower pressure) is used, water would flow into the cells, causing them to swell and potentially burst (hemolysis).
In this problem, we need to find the molar concentration of an NaCl solution that generates an osmotic pressure equal to that of the human body (7.8 bar).
Step 2: Key Formula or Approach:
We utilize the van't Hoff equation for osmotic pressure:
\[ \pi = i \cdot C \cdot R \cdot T \]
We are solving for the molar concentration (\(C\)).
Given data:
\(\pi = 7.8\) bar.
\(T = 37^{\circ}C = 37 + 273 = 310 K\) (Absolute temperature).
\(R = 0.083 L \cdot bar \cdot mol^{-1} \cdot K^{-1}\) (Gas constant in appropriate units).
For \(NaCl\), \(i = 2\) because it dissociates into \(Na^{+}\) and \(Cl^{-}\).
Step 3: Detailed Explanation:
Rearrange the equation to isolate the concentration \(C\):
\[ C = \frac{\pi}{i \cdot R \cdot T} \]
Substitute the known values into the equation:
\[ C = \frac{7.8}{2 \times 0.083 \times 310} \]
Let's compute the denominator first:
\[ 2 \times 0.083 = 0.166 \]
\[ 0.166 \times 310 = 51.46 \]
Now, calculate the final value of \(C\):
\[ C = \frac{7.8}{51.46} \]
\[ C \approx 0.15157... mol/L \]
By comparing this result with the given choices, the value closest to our calculation is \(0.15 mol/L\).
This result confirms the clinical fact that an approximately \(0.15 M\) solution of \(NaCl\) (which is roughly \(0.9% w/v\)) is physiological saline.
Step 4: Final Answer:
The required concentration of the NaCl solution is approximately 0.15 mol \(L^{-1}\).