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The atomic masses of the nuclides are given.

The Coulomb barrier height for alpha-particle emission,$U_2=30MeV$

Massive nuclides tend to undergo alpha decay releasing disintegration energy. The disintegration energy, also known as the Q-value, is the energy that is absorbed or released when a nuclear reaction takes place. The nuclear radii for both the system of decay are approximately in the same energy system, thus the potential barrier is only dependent on the product of charges.

Formulae:

The disintegration energy of a nuclear reaction,

$Q=\left(m_{parentnucleus}-\left(\underset{}{鈭�}{m}_{daughternuclei}\right)\right)c^2路路路路路路路路\left(1\right)$

The potential energy of a system of two charged nuclei, $U=\frac{k{q}_1{q}_2}{r}鈥�...\left(2\right)$

The disintegration energy or the Q-valuefor radium-223 鈥渄ecaying鈥� into lead-209 and carbon-12is given using the atomic masses and equation (1) as follows:

$$\begin{gathered} Q=\left(m_{{223}_{Ra}}-m_{{209}_{Pb}}-m_{{12}_C}\right)c^2 \\ =\left(223.01850u-208.98107u-14.00324u\right)\left(931.5MeV/u\right) \\ =31.8MeV \end{gathered}$$

Hence, the disintegration energy is $31.8MeV$.

The disintegration energy or the Q-valuefor radium-223 鈥渄ecaying鈥� into radium-219 and helium-4is given using the atomic masses and equation (1) as follows:

$$\begin{gathered} Q=\left(m_{{223}_{Ra}}-m_{{219}_{Rn}}-m_{4_{He}}\right)c^2 \\ =\left(223.01850u-219.00948u-4.00260u\right)\left(931.5MeV/u\right) \\ =5.98MeV \end{gathered}$$

Hence, the disintegration energy is 5.98 MeV.

From equation (2), we can get that the potential energy or the potential barrier height is proportional to$U伪\frac{q_1{q}_2}{r}$

Now, for the same universe that is the decay is from the same radium-223 nucleus, thus the nuclear radii of the system having the decay are approximately same.

Thus, using the above relation, the potential barrier height for the carbon-14 emission is given as follows:

$$\begin{gathered} U_1鈮�\left(\frac{q_{Pb}{q}_C}{q_{Rn}{q}_{He}}\right)U_2 \\ =\left(\frac{\left(82e\right)\left(6e\right)}{\left(86e\right)\left(2e\right)}\right)\left(30MeV\right) \\ =86MeV \end{gathered}$$

Hence, the value of the barrier height is 86 MeV.