Eleven molecules have speeds 15, 16, 17, …, 25 m/s. Calculate (a) v_avg and (b) v_rms.

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with calculating two quantities: (a) the average speed (vₐᵥ₉) and (b) the root mean square speed (vᵣₘₛ) for a set of eleven molecules with speeds ranging from 15 m/s to 25 m/s.

Step 2: To calculate vₐᵥ₉, use the formula for the average speed:

vₐᵥ₉ = (Σvᵢ)/N

, where

Σvᵢ

is the sum of all speeds and

N

is the total number of molecules. Add the speeds (15, 16, 17, ..., 25) and divide by 11.

Step 3: To calculate vᵣₘₛ, use the formula for the root mean square speed:

vᵣₘₛ = √((Σvᵢ²)/N)

, where

Σvᵢ²

is the sum of the squares of all speeds and

N

is the total number of molecules. Square each speed (15², 16², ..., 25²), sum them, divide by 11, and take the square root.

Step 4: Perform the summation for both calculations. For vₐᵥ₉, sum the speeds directly. For vᵣₘₛ, sum the squares of the speeds. Ensure you keep track of each step to avoid errors.

Step 5: Once the summations are complete, substitute the values into the respective formulas for vₐᵥ₉ and vᵣₘₛ. Simplify the expressions to find the numerical results for both quantities.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Average Speed (vₐᵥ₉)

Average speed is calculated by taking the total distance traveled divided by the total time taken. In the context of molecules, it can also be determined by summing the individual speeds of the molecules and dividing by the number of molecules. This provides a measure of the central tendency of the speeds in the group.

Root Mean Square Speed (vᵣₘₛ)

Root mean square speed is a statistical measure of the speed of particles in a gas. It is calculated by taking the square root of the average of the squares of the individual speeds. This concept is particularly useful in kinetic theory, as it relates to the temperature and energy of the gas molecules.

Kinetic Theory of Gases

The kinetic theory of gases explains the behavior of gases in terms of the motion of their molecules. It posits that gas molecules are in constant random motion and that their speeds contribute to the pressure and temperature of the gas. Understanding this theory is essential for interpreting the significance of average and root mean square speeds in the context of molecular motion.