Spectroscopic studies have shown that the fusion reaction in the sun's core proceeds by a complicated mechanism. One proposed overall equation for the solar fusion reaction is
411H+2 0−1e→42He
which is also accompanied by the release of "massless" neutrinos and photons in addition to energy. Precise masses of the reactants and product are given below.
Species 11H 0−1e 42He
Mass (amu) 1.007825 0.000549 4.002603
The sun produces energy via nuclear fusion at the rate of 4×1026J/s . Based on the proposed overall fusion equation, how long will the sun shine in years before it exhausts its hydrogen fuel? (Assume that there are 365 days in the average year.)
I am not sure how to start this problem
You can start by making the equation easier to read. In these situations it is easier to write the atomic number FIRST, then the symbol, then write the mass number SECOND so the equation would look like this.
4 1H1 + 2 -1e0 ==> 2He4 + heat + 0n0
4*mass 1H1 + 2*mass e - mass He = delta E in amu. Convert that to kg and use
delta E = delta M*c^2
That will give you the energy in joules per nuclear fusion.
To solve this problem, you need to find the mass of hydrogen fuel consumed per second and then divide the total mass of hydrogen in the sun by the rate of fuel consumption to determine how long it will last.
Let's begin by calculating the total mass of hydrogen in the sun. The atomic mass of hydrogen is 1.007825 amu per particle.
First, we need to convert the mass of the sun to amu. The mass of the sun is approximately 1.989 × 10^30 kg. Use Avogadro's number (6.022 × 10^23 particles/mole) to convert the mass to the number of particles.
1.989 × 10^30 kg × (1 mole / 1 g) × (6.022 × 10^23 particles/mole) = X particles
Now, we can calculate the total mass of hydrogen in the sun.
Mass of hydrogen = X particles × (1 amu / 6.022 × 10^23 particles) = Y amu
Next, we need to calculate the mass of hydrogen consumed per second.
The reaction equation states that 4 hydrogen nuclei (H) combine to form one helium nucleus (He). Therefore, the mass consumed per reaction is 4 x (1.007825 amu) = 4.0313 amu.
Now, using the rate of energy production, we can determine the rate of fuel consumption.
Energy produced per second = 4 × 10^26 J/s
Energy produced can be calculated using the equation
Energy produced = mass consumed per second × c^2
where c is the speed of light (approximately 3 × 10^8 m/s).
Substituting the known values,
mass consumed per second × (3 × 10^8 m/s)^2 = 4 × 10^26 J/s
Now, solve the above equation for mass consumed per second.
mass consumed per second = (4 × 10^26 J/s) / [(3 × 10^8 m/s)^2]
Finally, we can calculate the lifetime of the sun by dividing the total mass of hydrogen in the sun by the mass consumed per second.
Lifetime of the sun = Y amu / (mass consumed per second)
To convert this time to years, divide by the number of seconds in a year (365 days/year × 24 hours/day × 60 minutes/hour × 60 seconds/minute).
I hope this helps you to solve the problem!
To solve this problem, we need to calculate the total mass of hydrogen fuel in the Sun and then determine how much energy it produces per second. Finally, we can use the given rate of energy production to calculate how long the Sun will shine before it exhausts its hydrogen fuel.
Let's break down the steps:
Step 1: Calculate the number of hydrogen atoms (11H) in one amu (atomic mass unit):
- The mass of 1 amu is given as 1.007825 amu.
- So, the number of hydrogen atoms in 1 amu is 1/1.007825 ≈ 0.9921.
Step 2: Calculate the total mass of hydrogen fuel in the Sun:
- The mass of the hydrogen fuel is given as 1.007825 amu.
- Since 1 amu contains 0.9921 hydrogen atoms (from Step 1), the total number of hydrogen atoms in the Sun is: 1.007825 * 0.9921 = 0.9995.
- Therefore, the total mass of hydrogen fuel in the Sun is approximately 0.9995 times the mass of the Sun (since hydrogen makes up the majority of the Sun's mass).
Step 3: Calculate the energy produced per second:
- The rate of energy production is given as 4x10^26 J/s.
- This energy is produced by the fusion of 1 hydrogen atom (11H) into helium (42He) as given by the proposed overall equation.
- Therefore, the energy produced per fusion reaction can be calculated using the mass difference of the reactants and product:
Energy produced per fusion reaction = (Mass of the reactants - Mass of the product) x (speed of light)^2
= (1.007825 + 2*0.000549 - 4.002603) * (3x10^8 m/s)^2
Step 4: Calculate the total number of fusion reactions per second:
- Divide the rate of energy production (4x10^26 J/s) by the energy produced per fusion reaction (calculated in Step 3).
- This gives us the total number of fusion reactions happening per second in the Sun.
Step 5: Calculate the duration in years:
- Since we now know the total number of fusion reactions per second, we can calculate the duration in seconds before the hydrogen fuel is exhausted.
- Divide this duration in seconds by the number of seconds in a year (365 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute).
By following these steps, we can determine how long the Sun will shine before it exhausts its hydrogen fuel.