Here's how to estimate the lifetime of the sun based on the given information:

Energy from Hydrogen Fusion:

Every time two hydrogen atoms fuse, they release 2.3 × 10^-13 J of energy.

Mass of Converted Hydrogen:

To find the total mass of hydrogen converted per second to generate the sun's power output, we can divide the power (energy per second) by the energy released per fusion:

Mass of converted hydrogen per second (m_h_converted) = Sun's power / Energy per fusion

m_h_converted = (3.6 × 10^26 J/s) / (2.3 × 10^-13 J/fusion)

m_h_converted ≈ 1.57 × 10^39 kg/s (This is the mass of hydrogen converted to helium every second)

Mass of Fused Hydrogen Over Time:

We are estimating the lifetime (t) of the sun. To find the total mass of hydrogen converted over that time, we multiply the mass converted per second by the lifetime:

Total mass of converted hydrogen (M_h_converted) = m_h_converted * t

Relating Converted Hydrogen to Initial Mass:

We know the initial mass of the sun (m_sun) is 2 × 10^30 kg. Since we assumed the sun started as pure hydrogen, this initial mass represents the total amount of hydrogen available for fusion.

Not all the mass of the hydrogen atom is converted to energy during fusion. A small amount is converted to helium, which has a slightly higher mass.

To account for this, we can introduce a factor (f) representing the fraction of the hydrogen mass that gets converted to energy. This factor is less than 1 (around 0.007).

M_h_converted = f * m_sun

Solving for Lifetime:

Now we can equate the total mass of converted hydrogen over time (from step 3) to the initial mass of the sun (adjusted for conversion efficiency) from step 4:

m_h_converted * t = f * m_sun

(1.57 × 10^39 kg/s) * t = f * (2 × 10^30 kg)

t = (f * 2 × 10^30 kg) / (1.57 × 10^39 kg/s)

Finding the Lifetime:

f = 0.01 (1% conversion):

Converting Seconds to Years:

There are 31,536,000 seconds in a year. Converting the estimated lifetimes:

For f = 0.01: Lifetime ≈ 1.27 × 10^11 seconds / (31,536,000 seconds/year) ≈ 4.03 × 10^3 years (around 4,000 years)