You have a sample of gas at −33°C. You wish to increase the rms speed by a factor of 2. To what temperature should the gas be heated?

검증된 단계별 안내

insert step 1> Convert the initial temperature from Celsius to Kelvin using the formula: T(K) = T(°C) + 273.15.

insert step 2> Recall the formula for the root mean square (rms) speed of a gas: v_{rms} = \(\sqrt{\frac{3kT}{m}\)}, where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of a gas molecule.

insert step 3> Since you want to increase the rms speed by a factor of 2, set up the equation: 2v_{rms, initial} = v_{rms, final}.

insert step 4> Substitute the rms speed formula into the equation: 2\(\sqrt{\frac{3kT_{initial}\)}{m}} = \(\sqrt{\frac{3kT_{final}\)}{m}}.

insert step 5> Solve for T_{final} by squaring both sides and isolating T_{final}: T_{final} = 4T_{initial}.

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주요 개념

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RMS Speed of Gas

The root mean square (RMS) speed of gas molecules is a measure of the average speed of particles in a gas. It is directly related to the temperature of the gas, as higher temperatures result in greater kinetic energy and thus higher speeds. The formula for RMS speed is v_rms = √(3kT/m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of a gas particle.

Temperature and Kinetic Energy

Temperature is a measure of the average kinetic energy of the particles in a substance. In gases, as temperature increases, the kinetic energy of the molecules also increases, leading to higher speeds. This relationship is crucial for understanding how changes in temperature affect the motion of gas particles and their RMS speed.

Absolute Temperature Scale

The absolute temperature scale, measured in Kelvin (K), is essential for calculations in thermodynamics and gas laws. To convert Celsius to Kelvin, you add 273.15. Since the RMS speed is dependent on temperature in Kelvin, it is important to use this scale when determining the new temperature required to achieve a specific change in speed.