The density of carbon dioxide gas at `27^(@)C` and at pressure 1000 `N//m^(2)` is 1 kg `m^(-3)`. Find the root mean square speed of its molecule at `0^(@)C`. (pressure is constant)

Explanation

To find the root mean square speed (RMS speed) of carbon dioxide gas at \(0^\circ C\) given the density and pressure at \(27^\circ C\), we can follow these steps:

### Step 1: Understand the relationship between RMS speed, pressure, and density
The RMS speed \(v_{rms}\) of a gas can be expressed using the formula:
\[ v_{rms} = \sqrt{\frac{3P}{\rho}} \]
where \(P\) is the pressure and \(\rho\) is the density of the gas.

### Step 2: Substitute the known values
Given:
- Pressure \(P = 1000 \, N/m^2\)
- Density \(\rho = 1 \, kg/m^3\)

Substituting these values into the formula:
\[ v_{rms} = \sqrt{\frac{3 \times 1000}{1}} = \sqrt{3000} \]

### Step 3: Calculate the RMS speed at \(27^\circ C\)
Calculating \(\sqrt{3000}\):
\[ v_{rms} = \sqrt{3000} = 10\sqrt{30} \, m/s \]

### Step 4: Relate the RMS speeds at different temperatures
The RMS speed of a gas is proportional to the square root of its absolute temperature:
\[ \frac{v_{rms, 27}}{v_{rms, 0}} = \sqrt{\frac{T_{27}}{T_{0}}} \]
where:
- \(T_{27} = 27 + 273 = 300 \, K\)
- \(T_{0} = 0 + 273 = 273 \, K\)

### Step 5: Set up the equation
Substituting the temperatures into the equation:
\[ \frac{10\sqrt{30}}{v_{rms, 0}} = \sqrt{\frac{300}{273}} \]

### Step 6: Rearrange to find \(v_{rms, 0}\)
Rearranging gives:
\[ v_{rms, 0} = 10\sqrt{30} \cdot \sqrt{\frac{273}{300}} \]

### Step 7: Simplify the expression
Calculating the right-hand side:
\[ v_{rms, 0} = 10\sqrt{30} \cdot \sqrt{\frac{273}{300}} = 10\sqrt{30} \cdot \sqrt{\frac{91}{100}} = 10\sqrt{30} \cdot \frac{\sqrt{91}}{10} \]
Thus, we can simplify:
\[ v_{rms, 0} = \sqrt{30 \cdot 91} = \sqrt{2730} \]

### Step 8: Calculate the final value
Calculating \(\sqrt{2730}\):
\[ \sqrt{2730} \approx 52.25 \, m/s \]

### Final Answer
The root mean square speed of carbon dioxide molecules at \(0^\circ C\) is approximately \(52.25 \, m/s\).

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