To determine the speed of the resulting iridium isotope after a gold atom emits an alpha particle, we can apply conservation of momentum.
Let's denote the masses involved:
- The mass of the gold atom (\(^{197}\text{Au}\)) is approximately 197 atomic mass units (u).
- The mass of the alpha particle (\(^{4}\text{He}\)) is approximately 4 u.
- This means the mass of the resulting iridium isotope (\(^{193}\text{Ir}\)) is \(197 , \text{u} - 4 , \text{u} = 193 , \text{u}\).
Let:
- \(m_{\text{Au}} = 197 , \text{u}\)
- \(m_{\alpha} = 4 , \text{u}\)
- \(m_{\text{Ir}} = 193 , \text{u}\)
- \(v_{\alpha}\) = speed of the alpha particle
- \(v_{\text{Ir}}\) = speed of the iridium isotope
Before the emission, the gold atom is at rest, so its initial momentum is zero. After the emission, conservation of momentum states:
\[ 0 = m_{\alpha} v_{\alpha} + m_{\text{Ir}} v_{\text{Ir}} \]
Rearranging gives us:
\[ m_{\alpha} v_{\alpha} = - m_{\text{Ir}} v_{\text{Ir}} \] \[ v_{\alpha} = -\frac{m_{\text{Ir}}}{m_{\alpha}} v_{\text{Ir}} \]
If we also take account of the kinetic energy released (which is transformed during the alpha particle's emission), we can further explore this scenario. The kinetic energies of the alpha particle and the iridium can be expressed in terms of their momentum and mass, which can provide a relationship between their velocities.
To give an idea of their relative speeds, if we take typical values for the energy released during alpha decay (say around 5 MeV), we can estimate that the speed of the alpha particle will generally be much higher than that of the resulting iridium due to its smaller mass.
Using the above relations, and knowing typical velocities of alpha particles are usually around \(10^7 \text{ m/s}\) for many radioactive decays, and the fact that \(m_{\alpha} \ll m_{\text{Ir}}\), we can conclude that the speed of the iridium will be proportionally high but significantly less than that of the alpha particle.
In practical cases, you would sometimes find numerical solvers or further detail regarding the energy released, allowing you to specify exact numbers. However, conceptually, the iridium isotope will recoil slower than the alpha particle due to its greater mass.
If experimental values are needed (e.g., specific energy release levels or speeds), numerical calculation or experimental data would be necessary to specify a precise speed for the iridium isotope.