The molecules of a given massof a gas have RMS velocity of 200m/s at 27⁰ C and 10⁵ N/m² pressure .when the temperature is 127⁰ C and pressure 0.5 * 10⁵ N/m² ,the rms velocity in m/s is

To determine the root mean square (RMS) velocity of a gas under different conditions of temperature and pressure, we can use the relationship between these variables. The RMS velocity of a gas is influenced by both its temperature and pressure, and we can apply the ideal gas law along with the formula for RMS velocity to find the answer.

Understanding RMS Velocity

The RMS velocity (\(v_{rms}\)) of gas molecules is given by the formula:

v_rms = √(3RT/M)

Where:

• R is the universal gas constant (8.314 J/(mol·K)),
• T is the absolute temperature in Kelvin,
• M is the molar mass of the gas in kg/mol.

Converting Temperature to Kelvin

First, we need to convert the temperatures from Celsius to Kelvin:

• For 27 °C: \(T_1 = 27 + 273.15 = 300.15 \, K\)
• For 127 °C: \(T_2 = 127 + 273.15 = 400.15 \, K\)

Using the Ideal Gas Law

According to the ideal gas law, the relationship between pressure, volume, and temperature can be expressed as:

P_1/T_1 = P_2/T_2

Where \(P_1\) and \(P_2\) are the initial and final pressures, and \(T_1\) and \(T_2\) are the initial and final temperatures. We can rearrange this to find the new RMS velocity.

Calculating the New RMS Velocity

Given:

• Initial RMS velocity \(v_{rms1} = 200 \, m/s\)
• Initial pressure \(P_1 = 10^5 \, N/m^2\)
• Final pressure \(P_2 = 0.5 \times 10^5 \, N/m^2\)

Using the ratio of pressures and temperatures, we can express the new RMS velocity as:

v_rms2 = v_rms1 × √(P₂/P₁) × √(T₂/T₁)

Substituting Values

Now, substituting the known values:

• Pressure ratio: \(P_2/P_1 = 0.5 \times 10^5 / 10^5 = 0.5\)
• Temperature ratio: \(T_2/T_1 = 400.15 / 300.15\)

Calculating the temperature ratio:

T₂/T₁ ≈ 1.333

Final Calculation

Now we can plug these ratios into the RMS velocity equation:

v_rms2 = 200 × √(0.5) × √(1.333)

Calculating each part:

• √(0.5) ≈ 0.707
• √(1.333) ≈ 1.155

Now, multiplying these values:

v_rms2 ≈ 200 × 0.707 × 1.155 ≈ 163.5 \, m/s

Final Result

Thus, the RMS velocity of the gas at 127 °C and 0.5 × 10⁵ N/m² is approximately 163.5 m/s.