Question

How many coulombs of positive charge are there in 4.00 kg of plutonium, given its atomic mass is 244 and that each plutonium atom has 94 protons?

Short answer

4.00 kg of plutonium has a net charge of $1.48\times {10}^8\mathrm{C}$.

Step-by-step solution

Given Data

• Mass of plutonium (m) = 4.00 kg.
• Atomic mass of plutonium is 244.

• Each atom of plutonium has 94 protons.

Quantization of charge

According to the quantization of charge, every charged body is made of fundamental units of charge (electrons and protons). The net charge on the charged body equals the integral multiple of this fundamental charge.

Total number of protons

The number of moles of the plutonium atom is,

$N=\frac{m}{M}$

Here, N is the number of moles of the plutonium, m is the mass of the plutonium (m = 4.00 kg), and is the molar mass of the plutonium (M = 244 g).

Substituting all known values,

$$\begin{gathered} N=\frac{\left(4.00kg\right)}{\left(244g\right)} \\ =\frac{\left(4.00kg\right)\times \left({\displaystyle \frac{1000g}{1kg}}\right)}{\left(244g\right)} \\ =16.39 \end{gathered}$$

Since one mole of a substance contains $6.022\times {10}^{23}$elementary entities of the given substance. Therefore, the total number of atoms in 4.00 kg of plutonium is,

$$\begin{gathered} N_a=\left(16.39\right)\times \left(6.022\times {10}^{23}\right) \\ =9.87\times {10}^{24} \end{gathered}$$

We know that one atom of plutonium contains 94 protons. Therefore, the total number of protons in 4.00 kg of plutonium is,

$$\begin{gathered} n=\left(94\right)\times \left(9.87\times {10}^{24}\right) \\ =9.28\times {10}^{26} \end{gathered}$$

Net charge

According to the quantization of charge,

$Q=n\left|q_e\right|$

Here, Q is the net charge,n is the number of protons in 4.00 kg of plutonium$\left(n=9.28\times {10}^{26}\right)$ , and$\left|q_e\right|$ is the fundamental unit of charge $\left(\left|q_e\right|=1.6\times {10}^{-19}C\right)$.

Substituting all known values,

$$\begin{gathered} Q=\left(9.28\times {10}^{26}\right)\times \left(1.6\times {10}^{-19}C\right) \\ =1.48\times {10}^{8}C \end{gathered}$$

Hence, the net charge on 4.00 kg of plutonium is $1.48\times {10}^{8}C$.