Q.Calculate the mass of an $\alpha$-particle in atomic mass unit (u). Given: Mass of a normal helium atom $= 4.002603$ u; Mass of carbon atom $= 1.9926 \times 10^{-26}$ kg.

The α-particle is the nucleus of a helium-4 atom, so its mass equals the mass of a neutral helium atom minus the mass of its two electrons. The carbon-atom mass is given so that the electron mass (known in kg) can be converted into atomic mass units.

Step 1 – Value of 1 u in kg

By definition, 1 u is $\frac{1}{12}$ of the mass of a carbon-12 atom:

Step 2 – Mass of two electrons in u

With $m_e = 9.1 \times 10^{-31}$ kg:

$m_e = \frac{9.1 \times 10^{-31}}{1.6605 \times 10^{-27}} \approx 5.49 \times 10^{-4}\ \text{u}$ …

Concept & Intuition

An $\alpha$-particle is simply the nucleus of a helium-4 atom — two protons and two neutrons bound together. When we are given the mass of a neutral helium atom, that mass includes the two electrons orbiting the nucleus. To get the mass of just the nucleus (the $\alpha$-particle), we must subtract the mass of those two electrons.

The tricky part is that atomic mass units (u) are defined relative to carbon-12. The problem gives us the mass of a carbon atom in kilograms, which lets us find the conversion factor between kilograms and atomic mass units. Once we know how many kg equals 1 u, we can express the electron mass in u and do the subtraction cleanly.

Step-by-step solution

1. Find the mass of 1 atomic mass unit (u) in kg.

The definition: 1 u is exactly $1/12$ of the mass of one neutral carbon-12 atom. We are given the mass of one carbon atom as $1.9926 \times 10^{-26}$ kg.

So:

This is the standard conversion factor.

2. Find the mass of an electron in u.

The rest mass of an electron is a well-known constant: $m_e = 9.109 \times 10^{-31}$ kg. To convert this to u, divide by the value we just found: