Question

The average speed of a nitrogen molecule in air is proportional to the square root of the temperature in kelvins \((\mathrm{K}) .\) If the average speed is \(475 \mathrm{m} / \mathrm{s}\) on a warm summer day (temperature $=300.0 \mathrm{K}\( ), what is the average speed on a cold winter day \)(250.0 \mathrm{K}) ?$

Short answer

Answer: The average speed of a nitrogen molecule on a cold winter day is approximately 433.7 m/s.

Step-by-step solution

Write the proportionality equation

First, we need to write the proportionality equation relating the average speed of the nitrogen molecule (v) to the square root of the temperature in kelvins (T): v = k * sqrt{T} where k is the constant of proportionality.

Find the constant of proportionality (k) using given data

Using the given data: average speed of 475 m/s on a warm summer day (temperature = 300.0 K), we can find the constant of proportionality (k). Inserting the values into the equation gives: 475 = k * sqrt{300} To find k, first calculate the square root of 300: sqrt{300} ≈ 17.32 Next, solve for k by dividing both sides of the equation by 17.32: k ≈ 475 / 17.32 ≈ 27.42

Use the constant of proportionality (k) to find the average speed on a cold winter day

Now that we have found the constant of proportionality (k), we can use it to find the average speed on a cold winter day (temperature = 250.0 K). Insert the value of k and the temperature into the equation: v = 27.42 * sqrt{250} First, calculate the square root of 250: sqrt{250} ≈ 15.81 Next, multiply the result by the constant of proportionality (k) to find the average speed: v ≈ 27.42 * 15.81 ≈ 433.7 m/s So, the average speed of a nitrogen molecule on a cold winter day is approximately 433.7 m/s.