The number n of gas particles in a leak-tight container is approximately described by the equation , where T is the temperature of the gas in kelvins (K), P is the pressure of the gas in atmospheres (atm), and V is the volume of the container in liters (L). For a particular sample, an experimenter finds T = 296.6 $\pm$ 0.01 K, P = 1.05 $\pm$ 0.01 atm, and V = 1.302 $\pm$ 0.002 L. Use a linear approximation to estimate the range of the computed value of n.
A) 0 to
B) to
C) to
D) to

Question

The number n of gas particles in a leak-tight container is approximately described by the equation  , where T is the temperature of the gas in kelvins (K), P is the pressure of the gas in atmospheres (atm), and V is the volume of the container in liters (L). For a particular sample, an experimenter finds T = 296.6 $\pm$ 0.01 K, P = 1.05 $\pm$ 0.01 atm, and V = 1.302 $\pm$ 0.002 L. Use a linear approximation to estimate the range of the computed value of n.
A) 0 to 
B) to 
C) to 
D) to

Quizplus · Accepted Answer

Using the given equation, we can calculate the value of n as:

n = (P*V)/(R*T)

where R is the gas constant.

Taking partial derivatives of n with respect to each variable and using the given uncertainties, we can estimate the uncertainty in n using linear approximation:

Δn ≈ |∂n/∂P|ΔP + |∂n/∂V|ΔV + |∂n/∂T|ΔT

Δn ≈ (V/(R*T))ΔP + (P/(R*T))ΔV + (-P*V/(R*T^2))ΔT

Plugging in the given values:

Δn ≈ (1.302/(0.082057*296.6))*0.01 + (1.05/(0.082057*296.6))*0.002 + (-1.05*1.302/(0.082057*(296.6)^2))*0.01

Δn ≈ 1.068 * 10^20

So the range of n is approximately:

11.00 * 10^20 ± 1.07 * 10^20

which can be written as:

(10.93 * 10^20 to 11.07 * 10^20)

This range falls between the option B.

The Number N of Gas Particles in a Leak-Tight Container ±\pm±

The Number N of Gas Particles in a Leak-Tight Container $\pm$