Calculation of Heat Energy
When steam at 100 degree Celsius is passed into 20 g of water at 10 degree Celsius, the heat energy transferred can be calculated as follows:

Q = m_water × c_water × ΔT_water + m_steam × c_steam × ΔT_steam

where
m_water = mass of water = 20 g
c_water = specific heat capacity of water = 4.18 J/g°C
ΔT_water = change in temperature of water = 70°C (80°C - 10°C)
m_steam = mass of steam
c_steam = specific heat capacity of steam = 2.01 J/g°C
ΔT_steam = change in temperature of steam = -70°C (100°C - 30°C)

Assuming that all the steam condenses into water, we can calculate the mass of steam as follows:

Q = 20 g × 4.18 J/g°C × 70°C + m_steam × 2.01 J/g°C × (-70°C)
Q = 5,854 J - 141.7 m_steam
Q + 141.7 m_steam = 5,854 J
m_steam = (5,854 J - Q) / 141.7
m_steam = (5,854 J - 20 g × 4.18 J/g°C × 70°C) / 141.7
m_steam = 6.96 g

Therefore, the mass of steam is 6.96 g.

Calculation of Final Mass
The final mass of the water can be calculated as follows:

m_final = m_initial + m_steam
m_final = 20 g + 6.96 g
m_final = 26.96 g

However, we need to subtract the mass of the steam that condenses into water and then evaporates from the water. The heat energy required to evaporate this water can be calculated as follows:

Q = m_water × L_vaporization

where
L_vaporization = latent heat of vaporization of water = 2,260 J/g

Assuming that all the steam that condenses into water evaporates, we can calculate the mass of water that evaporates as follows:

Q = 6.96 g × 2,260 J/g
Q = 15,721.6 J

m_evaporated = Q / (c_water × ΔT_evaporated)
ΔT_evaporated = 100°C - 80°C = 20°C
m_evaporated = 15,721.6 J / (4.18 J/g°C × 20°C)
m_evaporated = 18.8 g

Therefore, the final mass of the water is:

m_final = m_initial + m_steam - m_evaporated
m_final = 20 g + 6.96 g - 18.8 g
m_final = 8.16 g

However, we need to consider that some of the steam may not have condensed into water due to heat losses to the surroundings. Assuming that 90% of the steam condenses into water, the final mass of the water is:

m_final = m_initial + 0.9 × m_steam - m_evaporated
m_final = 20