At what temperature will R.M.S speed of the molecules of the second member of the homologous series CnH2n+2 be the same as that of oxygen at 527°C

Engineering

Chemistry

Kinetic Theory of Gases KTG

Pressure Volume and Temperature

Ideal Gas Equation

Question

At what temperature will R.M.S speed of the molecules of the second member of the homologous series C_nH_2n+2 be the same as that of oxygen at 527°C

1227°C

580.3°C

750°C

477°C

Second member = C₂H₆ ; $\sqrt{\frac{3 {RT}_{1}}{M_{1}}} = \sqrt{\frac{3 {RT}_{2}}{M_{2}}}; \frac{T_{1}}{M_{1}} = \frac{T_{2}}{M_{2}}$
$\frac{T_{1}}{30} = \frac{800}{32}$ ⇒ T₁= 750 K ⇒ T₁ = 477°C

To solve this problem, we need to find the temperature at which the root mean square (R.M.S) speed of the molecules of the second member of the homologous series C_nH_2n+2 is the same as that of oxygen at 527°C.

Step 1: Identify the Second Member of the Homologous Series

The homologous series is C_nH_2n+2, which represents alkanes. The first member is methane (CH₄, n=1), the second member is ethane (C₂H₆, n=2). So, the gas in question is ethane (C₂H₆).

Step 2: Formula for R.M.S Speed

The R.M.S speed (v_rms) of a gas is given by:

v_{r m s} = \sqrt{\frac{3 R T}{M}}

where R is the universal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas in kg/mol.

Step 3: Set Up the Condition

We want the R.M.S speed of ethane to be equal to the R.M.S speed of oxygen at 527°C. Let:

For oxygen: T₁ = 527°C = 527 + 273 = 800 K, M₁ = 32 g/mol = 0.032 kg/mol
For ethane: T₂ = ? (in Kelvin), M₂ = (2*12 + 6*1) = 30 g/mol = 0.030 kg/mol

Set the R.M.S speeds equal:

\sqrt{\frac{3 R T_{1}}{M_{1}}} = \sqrt{\frac{3 R T_{2}}{M_{2}}}

Since the constants (3 and R) are the same, we can simplify to:

\frac{T_{1}}{M_{1}} = \frac{T_{2}}{M_{2}}

Step 4: Solve for T₂

Plug in the known values:

\frac{800}{0.032} = \frac{T_{2}}{0.030}

Calculate T₂:

T_{2} = 800 \times \frac{0.030}{0.032} = 800 \times 0.9375 = 750 K

Convert to Celsius: T₂ = 750 - 273 = 477°C.

Final Answer

The temperature is 477°C, which matches one of the options.

Detailed Solution

Step 1: Identify the Second Member of the Homologous Series

Step 2: Formula for R.M.S Speed

Step 3: Set Up the Condition

Step 4: Solve for T₂

Final Answer

Related Topics and Formulae

Detailed Solution

Step 1: Identify the Second Member of the Homologous Series

Step 2: Formula for R.M.S Speed

Step 3: Set Up the Condition

Step 4: Solve for T2

Final Answer

Related Topics and Formulae

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Step 4: Solve for T₂