To determine the speed of the resulting iridium isotope after a gold atom emits an alpha particle, we need to consider the principles of conservation of momentum and energy involved in the alpha decay process.
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Alpha Decay Process: When a gold atom (let's say \( ^{197}\text{Au} \)) undergoes alpha decay, it emits an alpha particle (\( ^{4}\text{He} \)). After emission, the gold atom transforms into an iridium isotope (in this case, \( ^{193}\text{Ir} \) because \( 197 - 4 = 193 \)).
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Momentum Conservation: The total momentum before decay must equal the total momentum after decay. If we denote:
- The mass of the gold atom before decay as \( m_{Au} \) (approximately \( 197 , \text{u} \)),
- The mass of the alpha particle as \( m_{\alpha} \) (approximately \( 4 , \text{u} \)),
- The mass of the resulting iridium isotope as \( m_{Ir} \) (approximately \( 193 , \text{u} \)),
- The velocity of the alpha particle after emission as \( v_{\alpha} \),
- The velocity of the resulting iridium isotope as \( v_{Ir} \).
Using conservation of momentum:
\[ 0 = m_{\alpha} v_{\alpha} + m_{Ir} v_{Ir} \]
Rearranging gives:
\[ m_{\alpha} v_{\alpha} = -m_{Ir} v_{Ir} \]
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Kinetic Energy Consideration: The kinetic energies of the alpha particle and the resulting iridium atom must also balance out (considering the energy released in the decay). The energy provided by the decay allows the alpha particle to have significant kinetic energy.
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Calculating Velocities: Without specific energy values or data from the decay of a particular gold isotope, we can't derive exact numerical speeds. However, typical speeds of alpha particles from decay can be in the range of \( 1 - 15 \) MeV, leading to velocities in the range of \( 10^6 \) m/s.
To summarize:
- The exact speed of the iridium isotope will depend on the details of the decay (energy released, etc.). However, you can use conservation of momentum to find the speed of the iridium after determining the speed of the alpha particle.
- The speed of the iridium will be inversely proportional to its mass compared to the alpha particle, meaning it will have lower speed than the alpha particle due to its higher mass.
If we assume, for example, that the alpha particle is moving at about \( 10^6 \) m/s, we would find \( v_{Ir} \) using the conservation of momentum equation above, where \( m_{Ir} \) is much larger than \( m_{\alpha} \):
\[ v_{Ir} \approx -\frac{m_{\alpha}}{m_{Ir}} v_{\alpha} \]
This will result in a much smaller velocity for the \( ^{193}\text{Ir} \) compared to the emitted alpha particle.