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The temperature at which oxygen molecules have the same root mean square speed as helium atoms have at \newline300K300\,\text{K} is :\newline(Atomic masses: \newlineHe=4u\text{He}=4\,\text{u},O=16u\text{O}=16\,\text{u} )\newline(11) \newline1200K1200\,\text{K}\newline(22) \newline600K600\,\text{K}\newline(33) \newline300K300\,\text{K}\newline(44) \newline2400K2400\,\text{K}

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Q. The temperature at which oxygen molecules have the same root mean square speed as helium atoms have at \newline300K300\,\text{K} is :\newline(Atomic masses: \newlineHe=4u\text{He}=4\,\text{u},O=16u\text{O}=16\,\text{u} )\newline(11) \newline1200K1200\,\text{K}\newline(22) \newline600K600\,\text{K}\newline(33) \newline300K300\,\text{K}\newline(44) \newline2400K2400\,\text{K}
  1. Formula Derivation: The root mean square speed (vrmsv_{\text{rms}}) of a gas is given by the formula vrms=3kTmv_{\text{rms}} = \sqrt{\frac{3kT}{m}}, where kk is the Boltzmann constant, TT is the temperature, and mm is the mass of a gas molecule. For two different gases to have the same vrmsv_{\text{rms}}, the temperatures must be inversely proportional to their molecular masses.
  2. Given Information: We are given that helium atoms have a vrmsv_{\text{rms}} at 300K300\,\text{K}. We need to find the temperature for oxygen molecules to have the same vrmsv_{\text{rms}}. Let's denote the temperature for oxygen as ToxygenT_{\text{oxygen}}.
  3. Equation Setup: Using the formula for vrmsv_{\text{rms}} and setting the vrmsv_{\text{rms}} for helium equal to the vrmsv_{\text{rms}} for oxygen, we get:\newline3kThelium/mhelium=3kToxygen/moxygen\sqrt{3kT_{\text{helium}}/m_{\text{helium}}} = \sqrt{3kT_{\text{oxygen}}/m_{\text{oxygen}}}\newlineSince the Boltzmann constant kk and the factor of 33 are common to both sides, they cancel out, leaving us with:\newlineThelium/mhelium=Toxygen/moxygen\sqrt{T_{\text{helium}}/m_{\text{helium}}} = \sqrt{T_{\text{oxygen}}/m_{\text{oxygen}}}
  4. Square Roots Elimination: We can now square both sides to get rid of the square roots: Theliummhelium=Toxygenmoxygen\frac{T_{\text{helium}}}{m_{\text{helium}}} = \frac{T_{\text{oxygen}}}{m_{\text{oxygen}}}
  5. Solving for ToxygenT_{\text{oxygen}}: We can rearrange the equation to solve for ToxygenT_{\text{oxygen}}:Toxygen=Thelium×(moxygenmhelium)T_{\text{oxygen}} = T_{\text{helium}} \times \left(\frac{m_{\text{oxygen}}}{m_{\text{helium}}}\right)
  6. Substitution and Calculation: Substitute the given values into the equation. We know that TheliumT_{\text{helium}} is 300K300\,\text{K}, mheliumm_{\text{helium}} is 4u4\,\text{u}, and moxygenm_{\text{oxygen}} is 16u16\,\text{u}:\newlineToxygen=300K×(16u4u)T_{\text{oxygen}} = 300\,\text{K} \times \left(\frac{16\,\text{u}}{4\,\text{u}}\right)
  7. Substitution and Calculation: Substitute the given values into the equation. We know that Thelium=300KT_{\text{helium}} = 300\,\text{K}, mhelium=4um_{\text{helium}} = 4\,\text{u}, and moxygen=16um_{\text{oxygen}} = 16\,\text{u}:\newlineToxygen=300K×(16u4u)T_{\text{oxygen}} = 300\,\text{K} \times \left(\frac{16\,\text{u}}{4\,\text{u}}\right) Perform the calculation:\newlineToxygen=300K×4T_{\text{oxygen}} = 300\,\text{K} \times 4\newlineToxygen=1200KT_{\text{oxygen}} = 1200\,\text{K}

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