3a. how do summer temperatures affect kinetic energy and the speed of methane molecules in the lake?

Explanation:

Step1: Recall kinetic - energy formula

The average kinetic energy of gas molecules is given by $K_{avg}=\frac{3}{2}kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature.

Step2: Analyze the effect of temperature on kinetic energy

In summer, the temperature $T$ increases. Since $K_{avg}\propto T$ (from the formula $K_{avg}=\frac{3}{2}kT$), as $T$ increases, the average kinetic energy of methane molecules in the lake increases.

Step3: Recall the relationship between kinetic energy and speed

The kinetic energy of a single molecule is $K = \frac{1}{2}mv^{2}$, where $m$ is the mass of the molecule and $v$ is its speed. If $K$ increases (as in the case of increased temperature) and the mass $m$ of the methane molecule is constant, from $K=\frac{1}{2}mv^{2}$, we can solve for $v=\sqrt{\frac{2K}{m}}$. So, as $K$ increases, the speed $v$ of the methane molecules increases.

Answer:

In summer, as the temperature increases, the kinetic energy of methane molecules in the lake increases because kinetic energy is directly proportional to temperature ($K_{avg}=\frac{3}{2}kT$). Since kinetic energy is related to speed by $K = \frac{1}{2}mv^{2}$, with the mass of methane molecules being constant, an increase in kinetic energy leads to an increase in the speed of methane molecules.