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 solve the problem of finding the root mean square speed of carbon dioxide molecules at \(0^\circ C\) given its density at \(27^\circ C\) and a pressure of \(1000 \, N/m^2\), we can follow these steps:

### Step 1: Understand the relationship between density, temperature, and pressure
We know from the ideal gas law that for a constant pressure, the density of a gas is inversely proportional to its temperature. This can be expressed as:
\[ \frac{D_1}{T_1} = \frac{D_2}{T_2} \]
where \(D_1\) and \(D_2\) are the densities at temperatures \(T_1\) and \(T_2\) respectively.

### Step 2: Convert temperatures to Kelvin
Given:
- \(T_1 = 27^\circ C = 27 + 273 = 300 \, K\)
- \(T_2 = 0^\circ C = 0 + 273 = 273 \, K\)

### Step 3: Use the density at \(27^\circ C\) to find the density at \(0^\circ C\)
Given that the density \(D_1\) at \(27^\circ C\) is \(1 \, kg/m^3\), we can find \(D_2\) (the density at \(0^\circ C\)):
\[ D_2 = D_1 \cdot \frac{T_2}{T_1} = 1 \cdot \frac{273}{300} \]
Calculating this gives:
\[ D_2 = \frac{273}{300} = 0.91 \, kg/m^3 \]

### Step 4: Use the formula for root mean square speed
The root mean square speed \(v_{rms}\) is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
However, we can also express it in terms of density:
\[ v_{rms} = \sqrt{\frac{P}{D}} \]
where \(P\) is the pressure and \(D\) is the density.

### Step 5: Substitute the known values into the equation
We know:
- \(P = 1000 \, N/m^2\)
- \(D = D_2 = 0.91 \, kg/m^3\)

Substituting these values into the equation gives:
\[ v_{rms} = \sqrt{\frac{1000}{0.91}} \]

### Step 6: Calculate the root mean square speed
Now we calculate:
\[ v_{rms} = \sqrt{1098.9011} \approx 33.12 \, m/s \]

### Final Answer
Thus, the root mean square speed of carbon dioxide molecules at \(0^\circ C\) is approximately \(33.12 \, m/s\).