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 = 292.9 $ \pm $ 0.01 K, P = 0.68 $ \pm $ 0.02 atm, and V = 1.731 $ \pm $ 0.005 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 = 292.9 $ \pm $ 0.01 K, P = 0.68 $ \pm $ 0.02 atm, and V = 1.731 $ \pm $ 0.005 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 linear approximation, we can estimate the range of the computed value of n by finding the derivative of n with respect to each variable (T, P, V) and then multiplying it with the corresponding error margin and adding up the results.

The derivative of n with respect to T is 11eaa948_cd98_20f4_84bc_b959f189cbb5_TB2342_11 P V, so the contribution of the error in T is approximately 0.042 x 11eaa948_cd98_20f5_84bc_eb58f11216f5_TB2342_11 P V = 9.520 x 10^17.

The derivative of n with respect to P is 11eaa948_cd98_20f6_84bc_12823a67edc3_TB2342_11 TV, so the contribution of the error in P is approximately 0.029 x 11eaa948_cd98_20f7_84bc_685535a7a0a4_TB2342_11 TV = 4.648 x 10^17.

The derivative of n with respect to V is 11eaa948_cd98_20f8_84bc_a35ad4dc9124_TB2342_11 TP, so the contribution of the error in V is approximately 0.0035 x 11eaa948_cd98_20f9_84bc_88d65509c5cc_TB2342_11 TP = 3.823 x 10^16.

Adding up the contributions, we get that the range of the computed value of n is approximately from 1.1311e25 - 9.520 x 10^17 - 4.648 x 10^17 - 3.823 x 10^16 to 1.1311e25 + 9.520 x 10^17 + 4.648 x 10^17 + 3.823 x 10^16, which simplifies to 1.1310e25 to 1.1312e25.

Therefore, the answer is B, 11eaa948_cd98_20f2_84bc_d19ff0d03a8f_TB2342_11 to 11eaa948_cd98_20f3_84bc_614cf3fdd79d_TB2342_11.

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

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