The average speed (in meters per second) of a gas molecule is vavg=√((8 R T)/(πM)) where T is th

step 1 All right, so taking the derivative with respect to T of 8R divided by pi m to the power of 1 ha

The average speed (in meters per second) of a gas molecule...

The average speed (in meters per second) of a gas molecule is $$ v_{\mathrm{avg}}=\sqrt{\frac{8 R T}{\pi M}} $$ where $T$ is the temperature (in kelvins), $M$ is the molar mass (in kilo-grams per mole), and $R=8.31 .$ Calculate $d v_{\text { avg }} / d T$ at $T=300 \mathrm{K}$ for oxygen, which has a molar mass of 0.032 $\mathrm{kg} / \mathrm{mol} .$

The average speed (in meters per second) of a gas molecule is
$$
v_{\mathrm{avg}}=\sqrt{\frac{8 R T}{\pi M}}
$$
where $T$ is the temperature (in kelvins), $M$ is the molar mass (in kilo-grams per mole), and $R=8.31 .$ Calculate $d v_{\text { avg }} / d T$ at $T=300 \mathrm{K}$ for oxygen, which has a molar mass of 0.032 $\mathrm{kg} / \mathrm{mol} .$

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Key Concepts

Gas Kinetic Theory

This theory explains the behavior of gas molecules in terms of their individual motions and collisions. It connects macroscopic properties such as temperature and pressure to microscopic properties like molecular speeds and energies, forming the basis for statistical treatments like the Maxwell-Boltzmann distribution.

Average Speed Formula

The average speed of gas molecules, as given by the formula v_avg = ?(8RT/(?M)), is derived from the Maxwell-Boltzmann distribution. It relates molecular speed to temperature and molar mass, reflecting how increasing temperature raises the kinetic energy and hence the speed of the molecules.

Function Differentiation

Differentiation is a fundamental calculus tool used to determine how a function changes as its input parameters change. In this context, finding d(v_avg)/dT involves calculating the derivative of the speed with respect to temperature to assess the sensitivity of molecular speed to temperature variations.

Chain Rule in Differentiation

The chain rule is used to differentiate composite functions. Since the average speed is defined as a square root of a function of temperature, the chain rule allows for the systematic differentiation of the outer square root function with respect to the inner function, which itself depends on temperature.

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