How long does it take electrons to get from the car battery to the starting motor? Assume the current is 137 A and the electrons travel through copper wire with cross sectional area 44.6 mm2 and length 75.7 cm. The mass density of copper is 8960 kg/m3 and the molar mass is 63.5 g/mol. Avogadro's number is 6.022 * 1023. Assume that each copper atom contributes one conduction electron. Answer in units of min.

t = 55.79 min

Explanation:

First, the problem is asking for calculate the time that it takes electrons from the battery to the motor.

The general formula to calculate time is:

t = d/V (1)

Where:

d: distance or length

V: speed

Now, we don't have data of speed, but we can know an expression of current density in function of the distance which is the following:

J = n*q*V (2)

Where:

q: charge of the particle (1.6x10^-19 C)

n: number of charge carriers per unit of volume

Current density (J) is actually current per Area so:

J = I/A (3)

Replacing (3) in (2) we have:

I/A = nqV

Solving for V:

V = I/Anq (4)

Finally, if we replace this expression in (1) we have:

t = nqAd / I (5)

Now, the value of n, it's not given but it can be calculated because we have mass density, molar mass and avogadro's number, so this value of "n" can be calculated using the following expression:

n = D * Av / MM (6)

Where:

D: mass density (kg/m³)

Av: avogadro number (6.02x10^23 atom/mol)

MM: molar mass (kg/mol)

Putting the data that we know to calculate n we have:

n = 8960 * 6.02x10^23 / 0.0635

n = 8.49x10^28 atom/m³

Now with the value of n, we can finally calculate the time:

t = nqAd / I

A is the area and it should be in m²: 44.6 mm² / 1x10^6 m = 4.46x10^-5 m²

d is the length in meter: 75.7 cm / 100 cm/m = 0.757 m

so replacing these data in (5):

t = 8.49x10^28 * 1.6x10^-19 * 4.46x10^-5 * 0.757 / 137

t = 3,347.63 s

But the answer is in minute so:

t = 3,347.63 / 60

t = 55.79 min

so the electrons takes 56 min aprox. to go from the car battery to the starting motor.