The time taken for the water to boil at the given rate of the alpha decay is 0.08 s.
The number of radium atom in 1.0 g is calculated as follows;
[tex]N = N_A(\frac{m}{M} )\\\\N = (6.02 \times 10^{23})(\frac{1}{223} )\\\\N = 2.7 \times 10^{21} \ atoms[/tex]
[tex]\lambda = \frac{ln2}{t_{1/2}} \\\\\lambda = \frac{ln 2}{(11.43 \times 86400)} \\\\\lambda = 7.02 \times 10^{-7} \ s^{-1}[/tex]
R = λN
R = (7.02 x 10⁻⁷) x (2.7 x 10²¹)
R = 1.895 x 10¹⁵ atoms/s
Δm = m(Ra₂₂₃) - [m(Rn₂₁₉) + m(α)]
Δm = 223.018u - (219.008u + 4.0026u)
Δm = 0.0074u
[tex]E =\Delta mc^2\\\\E = 0.0074u \times c^2\\\\E = 0.0074 \times931.5\ MeV/c^2 \times c^2\\\\E = 6.893 \ MeV\\\\E = 6.893 \times 10^6 \times 1.6\times 10^{-19} \\\\E= 1.103 \times 10^{-12} \ J[/tex]
P = E/t = ER
P = (1.103 x 10⁻¹²)(1.895 x 10¹⁵)
P = 2,089.99 J/s
mass = density x volume
mass = (1 g/L) x (0.46 L)
mass = 0.46 g
Q = mcΔT
Q = 0.46 x 4.184 x (100 - 16)
Q = 161.67 J
Pt = Q
t = Q/P
[tex]t = \frac{161.67}{2,089.99} \\\\t = 0.08 \ s[/tex]
Thus, the time taken for the water to boil at the given rate of the alpha decay is 0.08 s.
The complete question is below:
How long will it take the water to boil?
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