The temperature at which oxygen molecules have the same root mean square speed as helium atoms have at 300K is :(Atomic masses: He=4u,O=16u )(1) 1200K(2) 600K(3) 300K(4) 2400K
Q. The temperature at which oxygen molecules have the same root mean square speed as helium atoms have at 300K is :(Atomic masses: He=4u,O=16u )(1) 1200K(2) 600K(3) 300K(4) 2400K
Formula Derivation: The root mean square speed (vrms) of a gas is given by the formula vrms=m3kT, where k is the Boltzmann constant, T is the temperature, and m is the mass of a gas molecule. For two different gases to have the same vrms, the temperatures must be inversely proportional to their molecular masses.
Given Information: We are given that helium atoms have a vrms at 300K. We need to find the temperature for oxygen molecules to have the same vrms. Let's denote the temperature for oxygen as Toxygen.
Equation Setup: Using the formula for vrms and setting the vrms for helium equal to the vrms for oxygen, we get:3kThelium/mhelium=3kToxygen/moxygenSince the Boltzmann constant k and the factor of 3 are common to both sides, they cancel out, leaving us with:Thelium/mhelium=Toxygen/moxygen
Square Roots Elimination: We can now square both sides to get rid of the square roots: mheliumThelium=moxygenToxygen
Solving for Toxygen: We can rearrange the equation to solve for Toxygen:Toxygen=Thelium×(mheliummoxygen)
Substitution and Calculation: Substitute the given values into the equation. We know that Thelium is 300K, mhelium is 4u, and moxygen is 16u:Toxygen=300K×(4u16u)
Substitution and Calculation: Substitute the given values into the equation. We know that Thelium=300K, mhelium=4u, and moxygen=16u:Toxygen=300K×(4u16u) Perform the calculation:Toxygen=300K×4Toxygen=1200K
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