A plant virus is found to consist of uniform cylindrical particle of `150 Å` in diameter 5000 Å long. The specific volume of the virus is 0.75 `mLg^(-1)`. If the virus is considered to be a single particle, find its molar mass.

Explanation

To find the molar mass of the plant virus, we can follow these steps:

### Step 1: Calculate the Volume of the Virus
The virus is modeled as a cylinder. The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height (length) of the cylinder

Given:
- Diameter = 150 Å, so radius \( r = \frac{150}{2} = 75 \) Å
- Length \( h = 5000 \) Å

First, convert the dimensions from angstroms to centimeters:
- \( 1 \text{ Å} = 10^{-8} \text{ cm} \)
- Radius \( r = 75 \times 10^{-8} \text{ cm} \)
- Height \( h = 5000 \times 10^{-8} \text{ cm} \)

Now, calculate the volume:
\[ V = \pi (75 \times 10^{-8})^2 (5000 \times 10^{-8}) \]
\[ V = \pi (5.625 \times 10^{-15}) (5.0 \times 10^{-5}) \]
\[ V = \pi (2.8125 \times 10^{-20}) \text{ cm}^3 \]
\[ V \approx 8.83 \times 10^{-20} \text{ cm}^3 \]

### Step 2: Calculate the Mass of the Virus
Using the specific volume \( v = 0.75 \text{ mL/g} \) (which is equivalent to \( 0.75 \text{ cm}^3/\text{g} \)), we can find the mass \( m \) using the formula:
\[ m = \frac{V}{v} \]
Substituting the values:
\[ m = \frac{8.83 \times 10^{-20}}{0.75} \]
\[ m \approx 1.1773 \times 10^{-19} \text{ g} \]

### Step 3: Calculate the Molar Mass
To find the molar mass \( M \), we use Avogadro's number \( N_A = 6.022 \times 10^{23} \text{ mol}^{-1} \):
\[ M = m \times N_A \]
Substituting the values:
\[ M = (1.1773 \times 10^{-19}) \times (6.022 \times 10^{23}) \]
\[ M \approx 7.09 \times 10^{4} \text{ g/mol} \]

### Final Answer
The molar mass of the plant virus is approximately \( 7.09 \times 10^{4} \text{ g/mol} \).

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