Question

Using the approximation ∫v1+Δvv1f(v)dv≈f(v1)Δv for small Δv, estimate the fraction of nitrogen molecules at a temperature of 3.00×10² K that have speeds between 290 m/s and 291 m/s.

a) 0.01
b) 0.05
c) 0.10
d) 0.20

Answer 1

To estimate the fraction of nitrogen molecules at a temperature of 3.00 × 10² K that have speeds between 290 m/s and 291 m/s, we can use the given approximation ∫v1+Δvv1f(v)dv≈f(v1)Δv for small Δv.

Step-by-step explanation:

To estimate the fraction of nitrogen molecules at a temperature of 3.00 × 10² K that have speeds between 290 m/s and 291 m/s, we can use the given approximation ∫v1+Δvv1f(v)dv≈f(v1)Δv for small Δv.

The fraction can be estimated as:

Fraction = f(v1)Δv

To calculate this, we need to find the value of f(v1) for v1 = 290 m/s. The molecular speed distribution for nitrogen gas follows a Maxwell-Boltzmann distribution, which can be represented by the equation:

f(v) = 4π(v²/v_avg³) * e^((-v²)/v_avg²)

Where v_avg is the average speed of the molecules. We can plug in the appropriate values to calculate f(v1) and then multiply it by the small interval Δv = (291 - 290) m/s to estimate the fraction.