If the uncertainty in velocity and position of a minute particle in space are, 2.4 × 10⁻²⁶ (m s⁻¹) and 10⁻⁷ (m) respectively. The mass of the particle in g is ____________. (Nearest integer)
(Given : h = 6.626 × 10⁻³⁴ Js)

Solution

Chemistry ma’am · 4 steps

Put Δ p = m Δ v into Heisenberg's relation, solve for m, then turn kilograms into grams.

1. Heisenberg relation. Momentum is p = mv, so the uncertainty in momentum is Δ p = m Δ v. The mass is the only unknown, so rearrange the relation for m.

   m = h/(4π Δ x Δ v)

2. Put in numbers. The denominator is 4π × 10⁻⁷ × 2.4 × 10⁻²⁶ = 3.016 × 10⁻³². So m = (6.626 × 10⁻³⁴)/(3.016 × 10⁻³²) = 2.197 × 10⁻² kg.
3. Convert to grams. With h in J s and SI units for position and velocity, the mass comes out in kilograms. Multiply by 1000: 2.197 × 10⁻² kg = 21.97 g.
4. So. The mass of the particle is about 21.97 g, which rounds to 22.

The common slip: 44. Using h/2π in place of h/4π doubles the mass to about 43.9 g, so you get 44, not 22.

The idea to keep. Use Δ x · m Δ v = h/4π to find any one unknown. With SI values the mass comes out in kg, so convert if the question asks for grams.

Answer from the NTA final answer key. Solution by PrepWiser.